• Patients must have histologically or cytologically confirmed extensive stage SCLC (ES-SCLC) or other solid tumors for which carboplatin and etoposide are considered appropriate therapy

• No prior systemic therapy for extensive-stage, metastatic disease. Patients with prior limited stage disease who were treated with chemotherapy and concurrent radiation will be permitted to enroll as long as their previous treatment was 12 months or more prior to study enrollment

• Patients with treated brain metastases are eligible if they have stable symptoms and no ongoing requirement for corticosteroids as therapy for brain metastases

• Patients with untreated or progressive brain metastases (active brain metastases) are eligible if the treating physician determines that immediate CNS specific treatment is not required and is unlikely to be required during the first cycle of therapy. There must be no ongoing requirement for corticosteroids as therapy for brain metastases

• Previous radiation, including whole brain radiation, is allowed \>= 7 days of study registration. Stereotactic radiation therapy within 7 days is permitted

• Patients must have measurable disease by Response Evaluation Criteria in Solid Tumors (RECIST) version (v)1.1. At least one measurable lesion should be extra-cranial and outside of any portal of irradiation

• Archival tissue must be available, or patients must be willing to undergo a new biopsy to provide pre-treatment tumor sample (no intervening chemotherapy treatment, tissue must be from current extensive-stage/metastatic diagnosis)

• Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%)

• Absolute neutrophil count \>= 1,500/mcL

• Hemoglobin \>= 9.0 g/dL

• Platelets \>= 100,000/mcL

• International normalized ratio (INR) \< 1.5

• Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN). (This does not apply to patients with confirmed Gilbert's syndrome)

• Aspartate aminotransferase (AST)(serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT)(serum glutamate pyruvate transaminase \[SGPT\]) =\< 3 x institutional ULN, (if liver metastases present, can be up to 5 x ULN)

• Creatinine =\< institutional ULN OR glomerular filtration rate (GFR) \>= 60 mL/min/1.73 m\^2 (by the Cockcroft-Gault equation)

• Serum sodium \>= 130 mmol/L

• Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial

• 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

• Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, patients should be class II or better

• The effects of entinostat on the developing human fetus are unknown. For this reason and because histone deacetylase inhibitor (HDACi) 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, for the duration of study participation, up to 5 months after the last dose of study treatment. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, 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 5 months after completion of study treatment

• 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) and/or family member available will also be eligible