Sapanisertib, Carboplatin, and Paclitaxel in Treating Patients With Recurrent or Refractory Malignant Solid Tumors

Inclusion Criteria: * Patients must have a diagnosis of a solid tumor malignancy and is refractory to standard therapies who have relapsed after standard therapy, or whose cancers have no standard therapy. * Eastern Cooperative Oncology Group (ECOG) performance status and/or other performance status =\< 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 effective method of contraception and 1 additional effective (barrier) method, at the same time, from the time of signing the informed consent through 90 days (or longer as mandated by local labeling \[e.g., United Surgical Partners International (USPI), summary of product characteristics (SmPC), etc,\]) 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 patient (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.) * Male patients, even if surgically sterilized (i.e., status post-vasectomy), who: agree to practice highly effective barrier contraception during the entire study treatment period and through 120 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 patient. (Periodic abstinence \[e.g., calendar, ovulation, symptothermal, postovulation methods for the female partner\], 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 sperm during the course of this study or within 120 days after receiving their last dose of study drug. * Absolute neutrophil count (ANC) \>= 1.5 x 10\^9/L. * Platelet count \>= 100 x 10\^9/L. * Hemoglobin \>= 9 g/dL without transfusion within 1 week preceding study drug administration. * Total bilirubin =\< 1.5 x upper limit of normal (ULN). * Transaminases (aspartate aminotransferase/serum glutamic oxaloacetic transaminase \[AST/SGOT\] and alanine aminotransferase/serum glutamic pyruvic transaminase \[ALT/SGPT\]) =\< 2.5 x ULN (=\< 5 x ULN if liver metastases are present). * Creatinine clearance \>= 50 mL/min based either on Cockcroft-Gault estimate or based on urine collection (12 or 24 hour). * Glycosylated hemoglobin (HbA1c) \< 7%. * Fasting serum glucose =\< 130 mg/dL. * Fasting triglycerides =\< 300 mg/dL. * Ability to swallow oral medications. * 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 who have a history of brain metastasis are eligible for the study provided that all the following criteria are met: a) Brain metastases which have been treated b) No evidence of disease progression for \>= 3 months before the first dose of study drug. c) No hemorrhage after treatment d) Off-treatment with dexamethasone for 4 weeks before administration of the first dose of TAK-228 e) No ongoing requirement for dexamethasone or anti-epileptic drugs. * Patients must have evaluable or measurable disease by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 criteria. * Patients must be 4 weeks beyond previous treatment of any chemotherapy or radiotherapy, and must have recovered to =\< grade 1 or previous baseline for each toxicity. Exception: Patients may have received palliative low dose radiotherapy to the limbs 1-4 weeks before this therapy provided pelvis, sternum, scapulae, vertebrae, or skull were not included in the radiotherapy field. Patients who have received non-chemotherapeutic biological agents will need to wait at least 5 half-lives or 4 weeks, whichever is shorter, from the last day of treatment of non-chemotherapeutic biological agents. Exclusion Criteria: * Carboplatin or paclitaxel exposure within past 6 months. * Central nervous system (CNS) metastasis. * Other clinically significant co-morbidities, such as uncontrolled pulmonary disease, active central nervous system disease, active infection, or any other condition that could compromise the patient's participation in the study. * Known history of human immunodeficiency virus infection. * Known history of hepatitis B surface antigen-positive, or known or suspected active hepatitis C infection. * Any serious medical or psychiatric illness that could, in the investigator's opinion, potentially interfere with the completion of treatment according to this protocol. * Diagnosed or treated for another malignancy within 2 years before administration of the first dose of study drug, or previously diagnosed with another malignancy and have any evidence of residual disease. Patients with non-melanoma skin cancer or carcinoma in situ of any type are not excluded if they have undergone complete resection. * Breast feeding or pregnant. * Treatment with any investigational products within 4 weeks before the first dose of study drug. * Previous treatment with dual PI3K/mTOR inhibitors, TORC1/2 inhibitors or TORC1 inhibitors. * Manifestations of malabsorption due to prior gastrointestinal (GI) surgery, GI disease, or for an unknown reason that may alter the absorption of TAK-228. In addition, patients with enteric stomata are also excluded. * History of any of the following within the last 6 months before administration of the first dose of the drug: a) Ischemic myocardial event, including angina requiring therapy and artery revascularization procedures b) Ischemic cerebrovascular event, including transient ischemic attack and artery revascularization procedures c) Requirement for inotropic support (excluding digoxin) or serious (uncontrolled) cardiac arrhythmia (including atrial flutter/fibrillation, ventricular fibrillation or ventricular tachycardia) d) Placement of a pacemaker for control of rhythm e) New York Heart Association (NYHA) class III or IV heart failure f) Pulmonary embolism. * Significant active cardiovascular or pulmonary disease including: a) Uncontrolled hypertension (i.e., systolic blood pressure \> 150 mm Hg, diastolic blood pressure \> 90 mm Hg). Use of anti-hypertensive agents to control hypertension before cycle 1 day 1 is allowed. b) Pulmonary hypertension c) Uncontrolled asthma or oxygen (O2) saturation \< 90% by arterial blood gas analysis or pulse oximetry on room air d) Significant valvular disease; severe regurgitation or stenosis by imaging independent of symptom control with medical intervention, or history of valve replacement e) Medically significant (symptomatic) bradycardia f) History of arrhythmia requiring an implantable cardiac defibrillator g) Baseline prolongation of the rate-corrected QT interval (QTc) (e.g., repeated demonstration of QTc interval \> 480 milliseconds, or history of congenital long QT syndrome, or torsades de pointes). * Poorly controlled diabetes mellitus defined as glycosylated hemoglobin (HbA1c) \> 7%; patients with a history of transient glucose intolerance due to corticosteroid administration may be enrolled in this study if all other inclusion/exclusion criteria are met. * Patients receiving systemic corticosteroids (either IV or oral steroids, excluding inhalers or low-dose hormone replacement therapy) within 1 week before administration of the first dose of study drug. * Daily or chronic use of a proton pump inhibitor (PPI) and/or having taken a PPI within 7 days before receiving the first dose of study drug. * Patients with hypersensitivity or other allergic reaction to platinum chemotherapy. * Patients with hypersensitivity or other allergic reaction to taxanes.