Inclusion Criteria:

          -  Histologically and/or cytologically confirmed, non-small cell lung cancer (NSCLC) of
             adenocarcinoma histology at the time of initial diagnosis.

               -  Mixed tumors will be categorized by the predominant cell type; (Note: If small
                  cell elements are present the patient is ineligible)

               -  Known mutational status of KRAS and BRAF oncogenes.

               -  For patients in whom mutational testing result is unknown or unavailable from a
                  prior test, KRAS and BRAF testing will be performed (at a Clinical Laboratory
                  Improvement Act (CLIA) -certified laboratory) using an archived or fresh biopsy
                  as per standard of care, prior to enrollment.

          -  Stage IIIB-IV, locally advanced or metastatic disease according to the 7th edition of
             the American Joint Committee on Cancer (AJCC) lung cancer Tumor, Node, Metastasis
             (TNM) classification system

          -  Measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST)
             v1.1 criteria. Baseline measurements and evaluation of ALL sites of disease must be
             obtained within 4 weeks prior to enrollment.

          -  Failure of at least one line of systemic anti-cancer therapy for advanced NSCLC
             defined as either of the following:

               -  Radiological documentation of disease progression (or failure to achieve a
                  response) or

               -  Discontinuation due to toxicity

               -  Prior treatment with immunotherapy as well as maintenance therapy, including both
                  continuation and switch maintenance will be allowed if received at least 14 days
                  before start date of selumetinib-paclitaxel (immunotherapy is not allowed within
                  14 days of the start date of selumetinib-paclitaxel).

          -  Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1

          -  Estimated life expectancy, in the judgment of the investigator, which will permit
             receipt of treatment of 12 weeks or more

          -  Absolute neutrophil count >= 1.5x10^9 /L (1500 per mm^3)

          -  Platelets >= 100x10^9 /L (100,000 per mm^3)

          -  Hemoglobin > 9.0 g/dL

          -  Serum bilirubin < 1.5 x upper limit of normal (ULN) for institution [Exception:
             Patients who have elevated serum bilirubin due to underlying Gilbert's Syndrome or
             familial benign unconjugated hyperbilirubinemia are allowed.]

          -  Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase [SGOT]) <
             2.5 x upper limit of normal (ULN) for institution (or < 5.0 x ULN in presence of liver
             metastases)

          -  Alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase [SGPT]) < 2.5 x
             upper limit of normal (ULN) for institution (or < 5.0 x ULN in presence of liver
             metastases).

          -  Adequate renal function as defined by one of the following:

          -  Serum creatinine <= 1.5 mg/dl OR

          -  Serum creatinine clearance > 50ml/min (calculated by Cockcroft-Gault formula).

          -  Females of child bearing potential that are sexually active must agree to either
             practice 2 medically accepted highly effective methods of contraception at the same
             time or abstain from heterosexual intercourse from the time of signing the informed
             consent through 30 days after the last dose of study drug.

          -  Negative test for pregnancy is required for females of child-bearing potential; A
             female of child bearing potential is any woman, regardless of sexual orientation or
             whether they have undergone tubal ligation, who meets the following criteria:

               -  has not undergone a hysterectomy or bilateral oophorectomy; or

               -  has not been naturally postmenopausal for at least 24 consecutive months (i.e.,
                  has had menses at any time in the preceding 24 consecutive months or 730 days).

          -  Conception while on treatment must be avoided

          -  Males of child bearing potential must agree to practice effective barrier
             contraception during the entire study treatment period and through 6 months) after the
             last dose of study drug, (includes males surgically sterilized (i.e. status post
             vasectomy).

          -  Ability to understand and the willingness to sign a written informed consent and
             Health Insurance Portability and Accountability Act (HIPAA) document/s.

        Exclusion Criteria:

          -  Known actionable mutations (e.g., EGFR, ALK, ROS1), against which there is available
             treatment. Patients who progressed on such treatment, i.e., have developed acquired
             resistance and are no longer reasonably expected to derive therapeutic benefit are
             eligible for the trial.

          -  Any prior treatment with either a MEK, RAS, or RAF inhibitor for advanced or
             metastatic NSCLC.

          -  A history of hypersensitivity to selumetinib, or any excipient agents (e.g. Captisol
             or TPGS- a water soluble form of vitamin E)

          -  Any unresolved toxicity > Common Terminology Criteria for Adverse Events (CTCAE) grade
             2 despite optimal care/support, from previous anti-cancer therapy, except for
             alopecia, within 7 days prior to cycle 1, day 1.

          -  Cardiac conditions as follows:

               -  Uncontrolled hypertension (Blood Pressure (BP) >= 150/95 mmHg, despite medical
                  therapy)

               -  Left ventricular ejection fraction (LVEF) < 55%, measured by echocardiography

               -  Atrial fibrillation with a ventricular rate > 100 bpm on electrocardiogram (ECG)
                  at rest

               -  Symptomatic heart failure New York Heart Association (NYHA ) grade II-IV)

               -  Prior or current cardiomyopathy

               -  Severe valvular heart disease

               -  Uncontrolled angina (Canadian Cardiovascular Society grade II-IV despite medical
                  therapy)

               -  Acute coronary syndrome within 6 months prior to starting treatment

          -  Ophthalmological conditions as follows:

               -  Intra-ocular pressure > 21 mmHg, or uncontrolled glaucoma (irrespective of
                  intra-ocular pressure)

               -  Current or past history of central serous retinopathy or retinal vein occlusion.

          -  Prior treatment with chemotherapy or immunotherapy within 14 days prior to enrollment.
             Subjects receiving palliative radiation to central nervous system (CNS) disease within
             7 days may be eligible with PI approval. If the most recent treatment line is an
             EGFR-TKI, the washout period is a minimum of 3 days before the start of
             paclitaxel/selumetinib (i.e. treatment with the EGFR-TKI may continue until 3 days
             before start of study treatment). For other targeted therapy agents, the washout
             period will be 5 half-lives, prior to start of treatment on study.

          -  Caution should be exercised when administering paclitaxel (Taxol) concomitantly with
             medicines known to inhibit (e.g. ketoconazole and other imidazole antifungals,
             erythromycin, fluoxetine, gemfibrozil, cimetidine, ritonavir, saquinavir, indinavir
             and nelfinavir) OR induce (e.g. rifampicin, carbamazepine, phenytoin, efavirenz and
             nevirapine) either CYP2C8 or CYP3A4. Based on the in vitro data and SimCYP
             simulations, selumetinib is considered unlikely to perpetrate clinically significant
             drug-drug interaction via inhibition or induction of CYP enzymes.

          -  Major surgical procedure within 21 days prior to enrollment

          -  Brain metastases or spinal cord compression unless asymptomatic, treated and stable
             off steroids and anti-convulsants for at least 30 days

          -  Received an investigational drug within 30 days of starting treatment, or have not
             recovered from side effects of an investigational drug

          -  Female patients who are pregnant or breast-feeding (confirmation that the patient is
             not pregnant must be by a negative pregnancy test result obtained during screening;
             pregnancy testing is required of women of childbearing potential but not required for
             post-menopausal or surgically, sterilized women)

          -  Any evidence of severe or uncontrolled systemic disease, including, but not limited
             to, ongoing or active infection, active bleeding diatheses or renal transplant,
             including any patient known to have hepatitis B, hepatitis C, or human
             immunodeficiency virus (HIV).

          -  Serious medical or psychiatric illness/condition likely in the judgment of the
             investigator to interfere with compliance with protocol treatment/research