Regorafenib and Methotrexate in Treating Participants With Recurrent or Metastatic KRAS Mutated Non-Small Cell Lung Cancer

Inclusion Criteria: * Histologic or cytologic confirmed diagnosis of non-squamous non-small cell lung cancer that is recurrent or metastatic. * Documentation of pathogenic KRAS mutation * Previous receipt of at least one systemic therapy for recurrent or metastatic disease OR previous receipt of adjuvant systemic therapy within 6 months of enrollment; there is no limit on number of prior therapies allowed * Prior systemic therapy must be completed within 2 weeks of study treatment, with either improvement of clinically significant treatment-related toxicities to grade 0-1 OR stabilized to a new baseline * Previously treated OR asymptomatic non-progressing \< 1 cm untreated brain metastases are allowed * Measurable disease based on Response Evaluation Criteria in Solid Tumors (RECIST) version 1.1 criteria * Ability to understand and the willingness to sign a written informed consent document * Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 * Absolute neutrophil count (ANC) ≥ 1500/mm\^3 * Platelet count ≥ 100,000 /mm\^3 * Hemoglobin (Hb) ≥ 9 g/dL * Serum creatinine ≤ 1.5x upper limit of normal (ULN) OR calculated (Cockcroft Gault formula) or measured creatinine clearance ≥ 50 mL/min for patients with creatinine levels \> 1.5x ULN * Total bilirubin ≤ 1.5x ULN OR direct bilirubin ≤ ULN for patients with total bilirubin levels \> 1.5x ULN * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) ≤ 3x ULN (≤ 5x ULN for patients with liver involvement of their cancer) * Must be able to swallow and retain oral medication * Women patients of childbearing potential and men patients with women partners of childbearing potential must agree to use adequate contraception or agree to abstain from heterosexual activity beginning at the time of signing informed consent until at least 3 months after the last dose of study treatment; post-menopausal women (defined as no menses for at least 1 year) and surgically sterilized women are not considered childbearing Exclusion Criteria: * Previously treated with regorafenib * Known allergy to regorafenib or methotrexate * Currently receiving another systemic standard or investigational anti-cancer therapy; prior investigational therapy must be completed within 4 half-lives (if known) or 2 weeks, whichever is longer; the maximal washout of investigational therapy will not exceed 4 weeks prior to study treatment; bone medications such as bisphosphonates and receptor activator of nuclear factor kappa-Β (RANK) ligand inhibitors permitted * Leptomeningeal disease as documented by cerebrospinal fluid (CSF) cytology * Clinically significant cardiovascular related disease including: * Uncontrolled hypertension (systolic pressure \> 140 mm Hg or diastolic pressure \> 90 mmHg on repeated measurements, i.e., 3 or more separate days within one week) despite optimal medical management * Congestive heart failure - New York Heart Association (NYHA) class III or greater * Active coronary artery disease (i.e., unstable or new onset angina within 3 months of study treatment; myocardial infarction within 6 months of study treatment) * Clinically significant cardiac arrhythmias other than atrial flutter/fibrillation * Stroke, including transient ischemic attacks, within 6 months of study treatment * Other clinically significant arterial events, except for controlled asymptomatic pulmonary embolism, within 6 months of study treatment * Clinically significant hemorrhage or bleeding event within 1 month of study treatment * Uncontrolled symptomatic pleural effusion or ascites * Known active additional malignancy that is undergoing or expected to undergo systemic treatment during duration of study participation * Known history of human immunodeficiency virus (HIV) infection or known current active hepatitis B (i.e., hepatitis \[Hep\] B deoxyribonucleic acid \[DNA\] positive in prior 3 months) or hepatitis C infection (i.e., Hep C ribonucleic acid \[RNA\] positive in prior 3 months) * Major surgical procedure (e.g., involving the opening of a major body cavity) within 4 weeks of study treatment; this does not apply to low risk procedures (i.e., thoracentesis; paracentesis; chest tube / PleurX catheter placement; line placement; needle biopsy of tumor; and bronchoscopy) * Presence of a clinically significant non-healing wound, non-healing ulcer, or bone fracture * Concomitant therapy required at time of first dose of study treatment, including: * Strong CYP3A4 inhibitors and CYP3A4 inducers * Regular use of nonsteroidal anti-inflammatory drugs (NSAIDs), proton pump inhibitors, and probenecid * Women who are pregnant or breast feeding * Any condition which, in the investigator's opinion, including substance abuse, medical, psychological or social conditions that makes the patient unsuitable for trial participation or may interfere with the patient's participation in the study