M6620 and Carboplatin With or Without Docetaxel in Treating Patients With Metastatic Castration-Resistant Prostate Cancer

Inclusion Criteria: * Participants must have histologically or cytologically confirmed prostate cancer (code 10036910) with progressive disease at the time of study entry by either * Sequence of at least 2 rising PSA values at a minimum of 1-week intervals * Radiographic progression per RECIST1.1 for soft tissue and/or per PCWG3\^2 for bone, with or without PSA progression * Patients must have metastatic disease by bone scan or other nodal or visceral lesions on computed tomography (CT) or magnetic resonance imaging (MRI) and a castrate level of testosterone (\< 50 ng/dL) and evaluable for disease response by either * Baseline PSA \>= 2.0 ng/mL OR * Measurable disease per RECIST 1.1 * NOTE: Subjects must maintain a castrate state; if they have not had an orchiectomy, they must continue to receive luteinizing hormone-releasing hormone (LHRH) or gonadotropin-releasing hormone (GnRH) agonists or antagonists unless intolerant * At least 2 prior treatments for castration resistant prostate cancer as follows: * Past progression or intolerance to at least one secondary hormonal therapy (abiraterone, enzalutamide, galeterone, apalutamide, darolutamide, orteronel, seviteronel or equivalent) * Past progression or intolerance to taxane-based chemotherapy * Eastern Cooperative Oncology Group (ECOG) performance status =\< 1 (Karnofsky \>= 70%) * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,500/mcL * Hemoglobin \>= 9 g/dL (transfusion permitted) * Platelets …