Platinum and Taxane Chemo in Met Castration Resistant Prostate Cancer Patients With Alterations in DNA Damage Response Genes

Inclusion Criteria: * Histologic diagnosis of adenocarcinoma of the prostate. The presence of neuroendocrine or small cell carcinoma will be exclusionary * Prior treatment with any ARPI, such as abiraterone, enzalutamide, apalutamide, or darotlutamide, is required. * Participants must have recovered to ≤ grade 1 from all reversible toxicity related to prior systemic or radiation therapy, with the exception of chemotherapy induced alopecia and grade 2 peripheral neuropathy, and have adequate washout as follows: * Longest of one of the following: * Two weeks for oral therapies (such as abiraterone, apalutamide, enzalutamide, darolutamide and olaparib) * Standard cycle of standard IV or IM therapies (such as radium 223 or Lu-177-PSMA-617 * 2 weeks, 5 half lives, or standard cycle length (whichever is longer) for investigational agents * Previous major surgery is permitted provided that surgery occurred at least 28 days prior to participant enrollment and that wound healing has occurred * Prior external beam radiation is permitted provided a minimum of 7 days have elapsed between the last dose of radiation and date of enrollment * Radiologically documented presence of metastatic disease within 28 days prior to randomization * Disease progression after ARPI therapy as assessed by the investigator with at least one of the following: * PSA progression with a minimum of two rising PSA values at least 1 week apart, at least 25% and 2ug/L above baseline/nadir. * Radiographic progression of soft tissue disease by RECIST 1.1 criteria or bone metastases by PCWG3 criteria. * Medical or surgical castration with serum testosterone levels \<50ng/dL or \<1.7mmol/L * Qualifying Tier I or Tier II (clinically significant/likely clinically significant or pathogenic / likely pathogenic) germline or somatic alterations involving one or more of the following DDR genes: BRCA1, BRCA2, ATM, ATR, BRIP1, BARD1, CDK12, CHEK1, CHEK2, ERCC2, FANCA, FANCC, FANCD2, FANCL, PALB2, RAD51B, RAD51C, RAD51D, RAD54L. Monoallelic gene deletions in isolation will not be eligible. * Age 18 years or older. * Eastern Cooperative Oncology Group (ECOG) Performance Status of ≤ 2 * Participants must have adequate organ and marrow function measured within 14 days prior to enrolment * Life expectancy \> 12 weeks. * If the participant and the participant's partner are of childbearing potential, they must agree to use medically accepted methods of contraception (e.g., barrier methods, including male condom, female condom, or diaphragm with spermicidal gel) during the course of the study and for 6 months after the last dose of study drug. * Participant is able (i.e. sufficiently fluent) and willing to complete the quality of life and health utility questionnaires in either English or French * Participant consent must be appropriately obtained in accordance with applicable local and regulatory requirements. Each participant must sign a consent form prior to enrollment in the trial to document their willingness to participate. * Participants must be accessible for treatment and follow-up. Investigators must assure themselves the participants enrolled on this trial will be available for complete documentation of the treatment, adverse events, and follow-up. * In accordance with CCTG policy, protocol treatment is to begin within 2 working days after participant enrolment. Exclusion Criteria: * Received prior platinum chemotherapy (i.e. carboplatin, cisplatin, oxaliplatin, satraplatin) or prior taxane chemotherapy (docetaxel, paclitaxel, cabazitaxel) at any time for the treatment of prostate cancer with the exception of docetaxel for mCSPCor adjuvant therapy after curative intent treatment for localized prostate cancer, as long as it was no more than 6 cycles and at least 12 months have elapsed from their last treatment to the date of enrollment. If the participant received prior platinum or taxane chemotherapy for other cancers, then please contact the CCTG office to discuss this further with the trial team. * Active anticancer systemic therapy or investigational agents within 14 days of randomization. * Uncontrolled intercurrent illness including, but not limited to: active infection, symptomatic congestive heart failure (NYHA Class III or IV heart disease), unstable angina pectoris, cardiac arrhythmia, uncontrolled hypertension or psychiatric illness/social situations that would limit compliance with study requirements. * Participants with myelodysplastic syndrome/acute myeloid leukemia. * Malignancy within the previous 2-years with a \> 30% probability of recurrence within 12 months, with the exception of non-melanoma skin cancer, and in-situ or superficial bladder cancer. * Participants with known symptomatic brain metastasis. However, participants with asymptomatic, treated brain metastases that have been stable for at least 12 weeks are eligible for study entry. * Participants with symptomatic or impending cord compression unless appropriately treated beforehand and clinically stable and asymptomatic. * Presence of a condition or situation, which, in the investigator's opinion, may put the participant at significant risk, may confound the study results, or may interfere significantly with participation in the study. * Live attenuated vaccination administered within 30 days prior to randomization. * For patients with known immunodeficiency virus (HIV) infection, the HIV viral load must be undetectable on effective anti-retroviral therapy within 6 months of enrollment. * For participants with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated. Participants with a history of hepatitis C virus (HCV) infection must have been treated and cured. For participants with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load. * Unable to obtain provincial reimbursement of carboplatin and docetaxel