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 s…