Inclusion Criteria:

          -  Patients must have histologically or cytologically confirmed extensive stage SCLC
             (ES-SCLC) or other solid tumors for which carboplatin and etoposide are considered
             appropriate therapy

          -  No prior systemic therapy for extensive-stage, metastatic disease. Patients with prior
             limited stage disease who were treated with chemotherapy and concurrent radiation will
             be permitted to enroll as long as their previous treatment was 12 months or more prior
             to study enrollment

          -  Patients with treated brain metastases are eligible if they have stable symptoms and
             no ongoing requirement for corticosteroids as therapy for brain metastases

          -  Patients with untreated or progressive brain metastases (active brain metastases) are
             eligible if the treating physician determines that immediate CNS specific treatment is
             not required and is unlikely to be required during the first cycle of therapy. There
             must be no ongoing requirement for corticosteroids as therapy for brain metastases

          -  Previous radiation, including whole brain radiation, is allowed >= 7 days of study
             registration. Stereotactic radiation therapy within 7 days is permitted

          -  Patients must have measurable disease by Response Evaluation Criteria in Solid Tumors
             (RECIST) version (v)1.1. At least one measurable lesion should be extra-cranial and
             outside of any portal of irradiation

          -  Archival tissue must be available, or patients must be willing to undergo a new biopsy
             to provide pre-treatment tumor sample (no intervening chemotherapy treatment, tissue
             must be from current extensive-stage/metastatic diagnosis)

          -  Eastern Cooperative Oncology Group (ECOG) performance status =< 2 (Karnofsky >= 60%)

          -  Absolute neutrophil count >= 1,500/mcL

          -  Hemoglobin >= 9.0 g/dL

          -  Platelets >= 100,000/mcL

          -  International normalized ratio (INR) < 1.5

          -  Total bilirubin =< 1.5 x institutional upper limit of normal (ULN). (This does not
             apply to patients with confirmed Gilbert's syndrome)

          -  Aspartate aminotransferase (AST)(serum glutamic-oxaloacetic transaminase
             [SGOT])/alanine aminotransferase (ALT)(serum glutamate pyruvate transaminase [SGPT])
             =< 3 x institutional ULN, (if liver metastases present, can be up to 5 x ULN)

          -  Creatinine =< institutional ULN OR glomerular filtration rate (GFR) >= 60 mL/min/1.73
             m^2 (by the Cockcroft-Gault equation)

          -  Serum sodium >= 130 mmol/L

          -  Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral
             therapy with undetectable viral load within 6 months are eligible for this trial

          -  For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral
             load must be undetectable on suppressive therapy, if indicated

          -  Patients with a history of hepatitis C virus (HCV) infection must have been treated
             and cured. For patients with HCV infection who are currently on treatment, they are
             eligible if they have an undetectable HCV viral load

          -  Patients with a prior or concurrent malignancy whose natural history or treatment does
             not have the potential to interfere with the safety or efficacy assessment of the
             investigational regimen are eligible for this trial

          -  Patients with known history or current symptoms of cardiac disease, or history of
             treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac
             function using the New York Heart Association Functional Classification. To be
             eligible for this trial, patients should be class II or better

          -  The effects of entinostat on the developing human fetus are unknown. For this reason
             and because histone deacetylase inhibitor (HDACi) agents as well as other therapeutic
             agents used in this trial are known to be teratogenic, women of child-bearing
             potential and men must agree to use adequate contraception (hormonal or barrier method
             of birth control; abstinence) prior to study entry, for the duration of study
             participation, up to 5 months after the last dose of study treatment. Should a woman
             become pregnant or suspect she is pregnant while she or her partner is participating
             in this study, she should inform her treating physician immediately. Men treated or
             enrolled on this protocol must also agree to use adequate contraception prior to the
             study, for the duration of study participation, and 5 months after completion of study
             treatment

          -  Ability to understand and the willingness to sign a written informed consent document.
             Participants with impaired decision-making capacity (IDMC) who have a
             legally-authorized representative (LAR) and/or family member available will also be
             eligible

        Exclusion Criteria:

          -  Patients with evidence of leptomeningeal metastases (either by imaging or central
             nervous system [CNS] fluid findings)

          -  Patients who are receiving any other investigational agents

          -  History of allergic reactions attributed to compounds of similar chemical or biologic
             composition to entinostat or other agents used in study

          -  Patients with uncontrolled intercurrent illness

          -  Patients with psychiatric illness/social situations that would limit compliance with
             study requirements

          -  Pregnant women are excluded from this study because entinostat is HDACi agent with the
             potential for teratogenic or abortifacient effects and because of known teratogenic
             and abortifacient effects of cisplatin and etoposide. Because there is an unknown but
             potential risk for adverse events in nursing infants secondary to treatment of the
             mother with entinostat, and known risks with cisplatin and etoposide, breastfeeding
             should be discontinued if the mother is treated with entinostat. These potential risks
             may also apply to other agents used in this study

          -  Patients with a history of autoimmune disease (notable exceptions include
             hypothyroidism on thyroid replacement medication, type I diabetes, psoriasis or other
             cutaneous disease controlled with topical agents and without flare in 12 months
             requiring other treatment, celiac disease controlled with diet alone)

          -  Patients with a history of pulmonary fibrosis (history of radiation
             pneumonitis/fibrosis in the treatment field is permitted if stable and not requiring
             supplemental oxygen or corticosteroid use)

          -  Patients with prior history of allogeneic bone marrow or solid organ transplant

          -  Ongoing use of systemic corticosteroids or immunosuppressive agents within 14 days
             (inhaled corticosteroids, < 7 day course of prednisone for asthma/chronic obstructive
             pulmonary disease [COPD] exacerbation, or chronic low-dose supplemental steroids for
             adrenal insufficiency permitted)

          -  Prior treatment with anti-PD-1, or anti-PD-L1 therapeutic antibody or
             pathway-targeting agents.

               -  Patients who have received prior treatment with anti-CTLA-4 may be enrolled,
                  provided the following requirements are met:

                    -  Minimum of 12 weeks from the first dose of anti-CTLA-4 and > 6 weeks from
                       the last dose

                    -  No history of severe immune-related adverse effects from anti-CTLA-4
                       (National Cancer Institute [NCI] Common Terminology Criteria for Adverse
                       Events [CTCAE] grade 3 and 4)

          -  Treatment with systemic immunostimulatory agents (including, but not limited to,
             interferon [IFN]-alpha or interleukin [IL]-2) within 6 weeks prior to study
             registration

          -  Patients taking bisphosphonate therapy for symptomatic hypercalcemia. Use of
             bisphosphonate therapy for other reasons (e.g., bone metastasis or osteoporosis) is
             allowed

          -  Patients requiring treatment with a receptor activator of nuclear factor kappa-Β
             ligand (RANKL) inhibitor (e.g. denosumab) who cannot discontinue it before treatment
             with atezolizumab

          -  Patients requiring treatment with strong CYP3A inhibitors and inducers who cannot
             discontinue it before treatment with etoposide

               -  Because the list of these agents are constantly changing, it is important to
                  regularly consult a frequently updated list such as Facts and Comparisons or
                  Lexicompt; medical reference texts such as the Physicians' Desk Reference may
                  also provide this information

          -  Known hypersensitivity to Chinese hamster ovary cell products or other recombinant
             human antibodies

          -  History of severe allergic, anaphylactic, or other hypersensitivity reactions to
             chimeric or humanized antibodies or fusion proteins

          -  Known clinically significant liver disease, including active viral, alcoholic, or
             other hepatitis; cirrhosis; fatty liver; and inherited liver disease

          -  Patients with active tuberculosis (TB) are excluded

          -  Suspected or confirmed active severe acute respiratory syndrome coronavirus 2
             (SARS-CoV-2) infection (COVID-19).

               -  Those with a history of COVID-19 are eligible if they meet all of the above
                  eligibility criteria after clearance of COVID-19 by one of the following
                  criteria:

                    -  14 days have elapsed since symptom onset, the patient is afebrile, and
                       symptoms are improving for at least 72 hours

                    -  Have 2 negative specimens collected at least 24 hours apart

          -  Severe infections within 2 weeks prior to study registration, including, but not
             limited to, hospitalization for complications of infection, bacteremia, or severe
             pneumonia

          -  Received oral or intravenous (IV) antibiotics within 1 week prior to study
             registration. Patients receiving prophylactic antibiotics (e.g., for prevention of a
             urinary tract infection or chronic obstructive pulmonary disease) are eligible

          -  Major surgical procedure within 4 weeks prior to study registration or anticipation of
             need for a major surgical procedure during the course of the study. Common procedures
             such as biopsies, port insertions, and thoracenteses are allowed

          -  Administration of a live, attenuated vaccine within 4 weeks before study registration
             or anticipation that such a live, attenuated vaccine will be required during the study
             or up to 5 months after the last dose of atezolizumab