TAK-659 and Paclitaxel in Treating Patients With Advanced Solid Tumors

Inclusion Criteria: * All patients have advanced malignancy, including but not limited to high-grade epithelial ovarian cancer or cancer harboring K-RAS mutation, either refractory to standard therapy or for which no effective standard therapy is available * Patients must have measurable or evaluable disease, as defined by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 * Eastern Cooperative Oncology Group (ECOG) performance status 0 to 1 and life expectancy of greater than 3 months and/or other performance status of 0 to 1 * Female patients who: * Are postmenopausal for at least 1 year before the screening visit, OR * Are surgically sterile, OR * If they are of childbearing potential, agree to practice 1 highly effective method of contraception and one additional effective (barrier) method at the same time, from the time of signing the informed consent through 180 days after the last dose of study drug, OR * Agree to practice true abstinence, when is in line with the preferred and usual lifestyle of the subject. Periodic abstinence (e.g., calendar, ovulation, symptothermal, postovulation methods), withdrawal, spermicides only, and lactational amenorrhea are not acceptable methods of contraception. Female and male condoms should not be used together) * Agree not to donate eggs (ova) during the course of this study or 180 days after receiving their last dose of study drug * Male patients, even if surgically sterilized (i.e., status post-vasectomy), who: * Agree to practice effective barrier contraception during the entire study treatment period and through 180 days after the last dose of study drug, or * Agree to practice true abstinence, when this is in line with the preferred and usual lifestyle of the subject. (Periodic abstinence \[eg, calendar, ovulation, symptothermal, postovulation methods for the female partner\] and withdrawal are not acceptable methods of contraception. Female and male condoms should not be used together.) * Agree not to donate sperm during the course of this study or within 180 days after receiving their last dose of study drug * Absolute neutrophil count (ANC) \>= 1,500/uL * Platelet count \>= 100,000/uL (red blood cell count \[RBC\] and platelet transfusion allowed \>= 14 days before assessment) * Hemoglobin \>= 8 g/dL (RBC and platelet transfusion allowed \>= 14 days before assessment) * Total bilirubin =\< 1.5 x the upper limit of the normal range (ULN) (except patients with Gilbert's syndrome, who must have a total bilirubin =\< 3.0 mg/dL) * Aminotransferases alanine (ALT) and aspartate aminotransferase (AST) =\< 2.5 x ULN or =\< 5 x ULN if hepatic metastasis * Serum creatinine =\< 1 x ULN and creatine clearance \>= 60 mL/min either as estimated by the Cockcroft-Gault equation or based on urine collection (12 or 24 hours) if serum creatinine is abnormal * Lipase =\< 1.5 x ULN with no clinical symptoms suggestive of pancreatitis or cholecystitis * Amylase =\< 1.5 x ULN with no clinical symptoms suggestive of pancreatitis or cholecystitis * Blood pressure =\< grade 1 (hypertensive patients are permitted if their blood pressure is controlled to =\< grade 1 by hypertensive medications and glycosylated hemoglobin is =\< 6.5%) * Recovered (=\< grade 1 toxicity) from the reversible effects of prior anticancer therapy * Voluntary written consent must be given before performance of any study related procedure not part of standard medical care, with the understanding that consent may be withdrawn by the patient at any time without prejudice to future medical care * Patients may receive palliative radiation therapy immediately before or during the treatment if the radiation therapy is not delivered to the sole target lesions * Patients must be able to take oral medications without medical history of malabsorption or other chronic gastrointestinal disease, or conditions that may hamper compliance and/or absorption of the study agents * Patients have prior therapy with a taxane * Patients agree to provide archival tissue block or 10 formalin-fixed paraffin-embedded (FFPE) slides paraffin for use in pharmacodynamics correlative studies Exclusion Criteria: * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection requiring intravenous antibiotics, symptomatic congestive heart failure (New York Heart Association \[NYHA\] Class III or IV), or history of myocardial infarction, unstable angina, stroke or transient ischemic attack within 6 months prior to study enrollment * Active brain metastases or leptomeningeal metastases * Known hypersensitivity (eg, anaphylactic and anaphylactoid reactions) to any particular combination drug will result in a patient being ineligible for inclusion in that particular cohort * History of drug-induced pneumonitis requiring treatment with steroids; history of idiopathic pulmonary fibrosis, organizing pneumonia, or evidence of active pneumonitis on screening chest computed tomography (CT) scan; history of radiation pneumonitis in the radiation field (fibrosis) is permitted * Female patients who are both lactating and breastfeeding or have a positive serum pregnancy test during the screening period or a positive urine pregnancy test on day 1 before first dose of study drug * Any serious medical or psychiatric illness that could, in the investigator's opinion, potentially interfere with the completion of treatment according to this protocol * Life-threatening illness unrelated to cancer that could, in the investigator's opinion, make the patient not appropriate for this study * Known human immunodeficiency virus (HIV) positive * Known hepatitis B surface antigen positive, or known or suspected active hepatitis C infection * Any treatment specific for systemic tumor control within 3 weeks prior to the initiation of the study drugs; or within 2 weeks if cytotoxic agents were given weekly (within 6 weeks for nitrosoureas or mitomycin C), or within 5 half-lives for targeted agents with half-lives and pharmacodynamic effects lasting less than 4 days (that includes but is not limited to erlotinib, sorafenib, sunitinib, bortezomib, and other similar agents), or failure to recover from toxic effects of any therapy prior to the study drug treatment * Any clinically significant comorbidities, such as uncontrolled pulmonary disease, known impaired cardiac function or clinically significant cardiac disease (specified below), active central nervous system (CNS) disease, active infection, or any other condition that could compromise the patient's participation in the study. Patients with any of the following cardiovascular conditions are excluded: * Acute myocardial infarction within 6 months before starting study drug * Current or history of New York Heart Association Class III or IV heart failure * Evidence of current, uncontrolled cardiovascular conditions including cardiac arrhythmias, angina, pulmonary hypertension, or electrocardiographic evidence of acute ischemia or active conduction system abnormalities * Fridericia corrected QT interval (QTcF) \> 450 milliseconds (msec) (men) or \> 475 msec (women) on a 12-lead electrocardiogram (ECG) during the screening period * Abnormalities on 12-lead ECG including, but not limited to, changes in rhythm and intervals that, in the opinion of the investigator, are considered to be clinically significant * Use or consumption of any of the following substances: * Medications or supplements that are known to be inhibitors of P-glycoprotein (gp) and/or strong reversible inhibitors of CYP3A within 5 times the inhibitor half-life (if a reasonable half-life estimate is known), or within 7 days (if a reasonable half-life estimate is unknown), before the first dose of study drug. In general, the use of these agents is not permitted during the study except in cases where an adverse event (AE) must be managed * Medications or supplements that are known to be strong CYP3A mechanism-based inhibitors or strong CYP3A inducers and/or P-gp inducers within 7 days, or within 5 times the inhibitor or inducer half-life (whichever is longer), before the first dose of study drug. In general, the use of these agents is not permitted during the study except in cases where an AE must be managed * Grapefruit-containing food or beverages within 5 days before the first dose of study drug. Note that grapefruit-containing food and beverages are not permitted during the study * Major surgery within 14 days before the first dose of study drug and not recovered fully from any complications from surgery * Systemic infection requiring parenteral antibiotic therapy or other serious infection (bacterial, fungal or viral) within 21 days before the first dose of study drug. Patients who are at substantial risk of developing an infection may receive prophylaxis at the study of study treatment per investigator's discretion * Active secondary malignancy that requires treatment. Patients with non-melanoma skin cancer or carcinoma in situ of any type are not excluded if they have undergone complete resection and are considered disease-free at the time of study entry * Known gastrointestinal (GI) disease or GI procedure that could interfere with the oral absorption or tolerance of TAK-659 including difficulty swallowing tablets or diarrhea \> grade 1 despite supportive therapy. History of abdominal fistula, gastrointestinal perforation, or intra-abdominal abscess within 6 months prior to study enrollment * Patients with clinical bleeding, active gastric or duodenal ulcer * Grade 2 or higher peripheral neuropathy * Left ventricular ejection fraction (LVEF) is less than 50% on echocardiography (ECHO) or multiple-gated acquisition (MUGA) scanning or QTc is greater than 450 milliseconds (msec) for men and greater than 475 msec for women at screening * Treatment with high-dose corticosteroids for anticancer purposes within 14 days before the first dose of TAK-659; daily dose equivalent to 10 mg oral prednisone or less is permitted. Corticosteroids for topical use or in nasal spray or inhalers, or for premedication for paclitaxel are allowed * Lack of suitable venous access for the study-required blood sampling * Grade 2 or higher ophthalmologic disease