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