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

PRIMARY OBJECTIVES: I. To determine the progression free survival (PFS) of the combination of regorafenib and methotrexate for metastatic KRAS mutated non-small cell lung cancer (NSCLC) patients who have received at least 1 prior systemic therapy. SECONDARY OBJECTIVES: I. To determine the objective response rate (ORR) of the combination of regorafenib and methotrexate for metastatic KRAS mutated NSCLC patients who have received at least 1 prior systemic therapy. II. To determine the disease control rate (DCR) at 8 weeks of the combination of regorafenib and methotrexate for metastatic KRAS mutated NSCLC patients who have received at least 1 prior systemic therapy. III. To determine the safety of the combination of regorafenib and methotrexate in metastatic KRAS mutated NSCLC patients who have received at least 1 prior systemic therapy, assessed as the number of subjects that experience a treatment-emergent adverse event. IV. To determine the safety of the combination of regorafenib and methotrexate in metastatic KRAS-mutated NSCLC patients who have received at least 1 prior systemic therapy, assessed as the number (percent) of participants experiencing any dose-limiting toxicity (DLT). V. To determine the pharmacokinetic parameters of methotrexate when combined with regorafenib (i.e., trough and maximum serum concentration \[Cmax\]). OUTLINE: Participants receive regorafenib orally (PO) once daily (QD) and methotrexate PO twice weekly with 2-3 days apart on a 3 week on / 1 week off schedule. Courses repeat every 28 days in the absence of disease progression or unacceptable toxicity. After completion of study treatment, participants will come in for an end of study treatment visit.