Docetaxel and Carboplatin in Treating Patients With Metastatic, Castration Resistant Prostate Cancer Containing Inactivated Genes in the BRCA 1/2 Pathway

Inclusion Criteria: * Signed informed consent form (ICF) providing agreement to adhere to the dosing schedule, report for all trial visits and authorization, use and release of health and research trial information * Histologically or cytologically confirmed carcinoma of the prostate (excluding neuroendocrine differentiation or squamous cell histology) * Ongoing gonadal androgen deprivation therapy with gonadotropin-releasing hormone (GnRH) analogues, antagonists or orchiectomy; patients who have not had an orchiectomy must be maintained on effective GnRH analogue/antagonist therapy * Castration resistant prostate cancer as defined by rising PSA when serum testosterone \< 50 ng/ml (note: current testosterone results are not required if the potential subject has not missed any GnRH analogue/antagonist doses since their last result was received) AND one of the following: * PSA level of at least 2 ng/ml that has risen on at least 2 successive occasions at least 1 week apart * Evaluable disease progression by modified RECIST (Response Evaluation Criteria in Solid Tumors) * Progression of metastatic bone disease on bone scan with \> 2 new lesions * Prior therapy with abiraterone, enzalutamide and/or docetaxel; there is no limit to the number of prior treatment regimens * Presence of metastatic disease on scans * Eastern Cooperative Oncology Group (ECOG) performance status of =\< 1 * Life expectancy \>= 12 weeks * No prior malignancy is allowed except: * Adequately treat…