Selumetinib and Paclitaxel as Second-Line Treatment in Treating Patients With Stage IIIB-IV Non-small Cell Lung Cancer

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. * 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