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 treated basal cell or squamous cell skin cancer or * In situ carcinoma of any site or * Other adequately treated malignancy for which the patient has been disease-free for at least one year (any prior chemotherapy is allowed) * Absolute neutrophil count \>= 1.5 x 10\^9 cells/L * Hemoglobin (Hgb) \>= 9.0 g/dL * Platelets \>= 100,000 x 10\^9/L * Aspartate aminotransferase (AST), alanine aminotransferase (ALT) =\< 1.5 x upper limit of normal (ULN) * Total bilirubin levels =\< 1.5 x ULN * Patients must have clear and documented evidence of biallelic inactivation BRCA1, BRCA2 or ATM by sequencing, for example University of Washington (UW)-Oncoplex, SU2C, or Foundation One testing and/or patients with clearly deleterious mutations of other genes involved in homologous DNA repair (e.g., partner and localizer of BRCA2 \[PALB2\], BRCA1-interacting protein 1 \[BRIP1\], etc.) by sequencing via UW-Oncoplex, SU2C, or Foundation One testing may be included at the investigator's discretion Exclusion Criteria: * Currently receiving active therapy for other neoplastic disorders * Histologic evidence of neuroendocrine or small cell carcinoma of the prostate * Known parenchymal brain metastasis * Active or symptomatic viral hepatitis or chronic liver disease * Clinically significant heart disease as evidenced by myocardial infarction, or arterial thrombotic events in the past 6 months, severe or unstable angina, or New York Heart Association (NYHA) class II-IV heart disease or cardiac ejection fraction measurement of \< 35 % at baseline, if done * Treatment with an investigational therapeutic within 30 days of cycle 1 * Patients with dementia/psychiatric illness/social situations limiting compliance with study requirements or understanding and/or giving of informed consent are not eligible * Any medical conditions, which, in the opinion of the investigators, would jeopardize either the patient or the integrity of the data obtained are not eligible