Testing the Combination of MLN4924 (Pevonedistat), Carboplatin, and Paclitaxel in Patients With Advanced Non-small Cell Lung Cancer (NSCLC) Who Have Previously Been Treated With Immunotherapy

Inclusion Criteria: * Patients must be \>= 18 years old. Because no dosing or adverse event (AE) data are currently available on the use of MLN4924 (pevonedistat) in combination with carboplatin and paclitaxel in patients \< 18 years of age, children are excluded from this study, but may be eligible for future pediatric trials. * Patients must have histologically confirmed stage IIIB or IV NSCLC (squamous or nonsquamous) that is metastatic or unresectable. * Patients must have measurable disease per Response Evaluation Criteria in Solid Tumors (RECIST) 1.1. * Patients must have progressed on prior treatment with checkpoint inhibitor (PD-1/PD-L1 inhibitors) either as a single-agent therapy or in combination, as below. Patients will be eligible if there is a contra-indication to checkpoint inhibitor therapy. * Patients who have progressed after receiving a checkpoint inhibitor in combination with a platinum-based doublet, as first-line treatment for NSCLC. * Patients who have progressed on checkpoint inhibitor as second-line therapy, after receiving a platinum-based doublet as first-line therapy. * Patients who have progressed on platinum-based doublet as second-line therapy, after receiving a checkpoint inhibitor as first-line therapy. * Patients must have disease progression on or after platinum-based chemotherapy for metastatic disease or within 6 months of completion of platinum-based chemotherapy administration as adjuvant/neoadjuvant chemotherapy or concurrent chemoradiation. * Absolute neutrophil count \>= 1,500/mcL. * Platelet count \>= 150,000/mcL. * Total bilirubin =\< 1 x institutional upper limit of normal (ULN). Patients with Gilbert's syndrome may enroll if direct bilirubin =\< 1.5 x institutional ULN. * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\]) / alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3.0 x institutional ULN. Patients with metastatic liver disease may enroll if =\< 5 x ULN. * Glomerular filtration rate (GFR) \> 30 mL/min/1.73 m\^2. * Known human immunodeficiency virus (HIV) positive patients who meet the following criteria will be considered eligible: * CD4 count \> 350 cells/mm\^3. * Undetectable viral load within the last six months. * HIV positive patients must be stable on highly active antiretroviral therapy (HAART). Clinically significant metabolic enzyme inducers are not permitted during this study (e.g., ritonavir, efavirenz, nevirapine). * No history of acquired immune deficiency syndrome (AIDS)-defining opportunistic infections. * Eastern Cooperative Oncology Group (ECOG) performance status 0-1. * Life expectancy \>= 12 weeks. * Patients are eligible if central nervous system (CNS) metastases are adequately treated and neurological symptoms have returned to baseline or are controlled for at least 2 weeks prior to enrollment. In addition, subjects must be either off corticosteroids, or on stable or decreasing dose of steroids. Patients with leptomeningeal disease are excluded. * The effects of MLN4924 (pevonedistat) on the developing human fetus are unknown. For this reason and because agents as well as other therapeutic agents used in this trial are known to be teratogenic, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study or within 4 months of completion, she should inform her treating physician immediately. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of MLN4924 (pevonedistat) administration. 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 and 1 additional effective (barrier) method of contraception, at the same time, from the time of signing the informed consent through 4 months after the last dose of study drug (female and male condoms should not be used together), or * Agree to practice true abstinence, when this 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.) Male patients, even if surgically sterilized (i.e., status postvasectomy), who * Agree to practice effective barrier contraception during the entire study treatment period and through 4 months after the last dose of study drug (female and male condoms should not be used together), OR * Agree to practice true abstinence, when this is in line with the preferred and usual lifestyle of the subject. (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.) * For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated. * Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load. * Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial. * Left ventricular ejection fraction (LVEF) \>= 50% as assessed by echocardiogram or radionuclide angiography. * Ability to understand and the willingness to sign a written informed consent document. Participants with impaired decision-making capacity (IDMC) who have a legally authorized representative (LAR) or caregiver and/or family member available will also be considered eligible. * Patients with NSCLC harboring genomic aberrations (e.g. sensitizing EGFR, ALK, ROS1, NTRK, BRAF V600E mutation positive) must have received prior treatment with Food and Drug Administration (FDA) approved targeted therapy for patients for which FDA approved targeted therapies is available. Exclusion Criteria: * Patients who have not recovered from AEs due to prior anti-cancer therapy (i.e., have residual toxicities \> grade 1) with the exception of alopecia and neuropathy. * Patients who are receiving any other investigational agents. * History of allergic reactions attributed to compounds of similar chemical or biologic composition to MLN4924 (pevonedistat), carboplatin, or paclitaxel. * Patients who have had chemotherapy or radiotherapy within 2 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study. * Patients with uncontrolled intercurrent illness. * Patients with psychiatric illness/social situations that would limit compliance with study requirements. * Pregnant women are excluded from this study because MLN4924 (pevonedistat), carboplatin, and paclitaxel have the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with MLN4924 (pevonedistat), breastfeeding must be discontinued if the mother is treated with MLN4924 (pevonedistat). These potential risks may also apply to other agents used in this study. * Female patients who intend to donate eggs (ova) during the course of this study or 4 months after receiving their last dose of study drug(s). * Male patients who intend to donate sperm during the course of this study or 4 months after receiving their last dose of study drug(s). * Known cardiopulmonary disease defined as: * Unstable angina; * Congestive heart failure (New York Heart Association \[NYHA\] class III or IV;); * Myocardial infarction within 6 months prior to first dose (patients who had ischemic heart disease such as acute coronary syndrome \[ACS\], myocardial infarction, and/or revascularization greater than 6 months before screening and who are without cardiac symptoms may enroll); * Symptomatic cardiomyopathy; * Clinically significant arrhythmia: * History of polymorphic ventricular fibrillation or torsade de pointes, * Permanent atrial fibrillation, defined as continuous atrial fibrillation for \>= 6 months, * Persistent atrial fibrillation, defined as sustained atrial fibrillation lasting \> 7 days and/or requiring cardioversion in the 4 weeks before screening, * Grade 3 atrial fibrillation defined as symptomatic and incompletely controlled medically, or controlled with device (e.g., pacemaker), or ablation in the past 6 months and * Patients with paroxysmal atrial fibrillation or grade \< 3 atrial fibrillation for period of at least 6 months are permitted to enroll provided that their rate is controlled on a stable regimen. * Clinically symptomatic pulmonary hypertension requiring pharmacologic therapy. * Peripheral neuropathy that is grade \>= 3, or grade 2 with pain on clinical examination during the screening period. * Uncontrolled high blood pressure (i.e., systolic blood pressure \> 180 mmHg, diastolic blood pressure \> 95 mmHg). * Prolonged rate corrected QT (QTc) interval \>= 500 msec, calculated according to institutional guidelines. * Known moderate to severe chronic obstructive pulmonary disease, interstitial lung disease, and pulmonary fibrosis. * Major surgery within 14 days before the first dose of any study drug or a scheduled surgery during study period. * Life-threatening illness unrelated to cancer. * Patients with uncontrolled coagulopathy or bleeding disorder. * Known hepatic cirrhosis or severe pre-existing hepatic impairment.