Testing the Addition of an Anti-cancer Drug, Entinostat, to the Usual Chemotherapy and Immunotherapy Treatment (Atezolizumab, Carboplatin and Etoposide) for Previously Untreated Aggressive Lung Cancer That Has Spread

Inclusion Criteria: * 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 interv…