Testing Whether the Addition of Carboplatin Chemotherapy to Cabazitaxel Chemotherapy Will Improve Outcomes Compared to Cabazitaxel Alone in People With Castrate-Resistant Prostate Cancer That Has Spread Beyond the Prostate to Other Parts of the Body

Inclusion Criteria: * STEP 1 SCREENING REGISTRATION: NOTE: All participants must have biopsy tissue submitted to MD Anderson Cancer Center prior to randomization for alteration assessment. Participants must have determination of their AVPC-Molecular Pathologic Signature immunohistochemistry (MSIHC) status from central assessment by the MD Anderson Clinical Pathology Laboratory using Clinical Laboratory Improvement Act (CLIA) certified immunohistochemistry (IHC) assays for TP53, RB1 and PTEN. In addition, while not mandated, CLIA certified next generation sequencing (NGS) of tumor deoxyribonucleic acid (DNA) and/or circulating tumor derived DNA (ctDNA) assessment of AVPC-MS marker status will be collected from participants for whom it is available * STEP 1 SCREENING REGISTRATION: Participants must have a histologically confirmed diagnosis of prostate cancer at the time of step 1 registration * STEP 1 SCREENING REGISTRATION: Participants must have castrate-resistant prostate cancer and metastatic disease by bone scan and/or CT/MRI (i.e., soft tissue, visceral, lymph node) * STEP 1 SCREENING REGISTRATION: Participants may have received any prior therapy, but one must be docetaxel or contain docetaxel in either the castrate-sensitive and/or castrate resistant disease state * STEP 1 SCREENING REGISTRATION: Participants must be ≥ 18 years of age at the time of step 1 screening registration * STEP 1 SCREENING REGISTRATION: Participants must have solid tumor biopsy material (formalin-fixed paraffin-embedded (FFPE) tissue blocks and/or 10 cut slides on four-micron thick unstained positive charged slides of FFPE tissue) available for submission for alterations in TP53, RB1 and PTEN by IHC using CLIA certified assays in the MD Anderson Clinical Pathology Laboratory. This specimen is required for central assessment of the AVPC-MSIHC regardless of whether the site has already locally evaluated the AVPC-MS status * STEP 1 SCREENING REGISTRATION: Tumor samples submitted for analysis must have been collected within 12 months prior to step 1 screening registration. Samples from metastatic lesions collected in the castrate-resistant disease state are preferable but not mandatory. Samples obtained during the hormone-naive disease state are acceptable if collected within 12 months of step 1 screening registration. If more than one tumor sample exists, the sample obtained closest to the date of registration should be submitted to MDACC for analysis * NOTE: Sites will receive an email from Southwest Oncology Group (SWOG) Statistics and Data Management Center containing participant results of Aggressive Variant Prostate Cancer Molecular Signature (AVPC-MS) assessment within 5-12 business days after tissue submission to MD Anderson Clinical Pathology Laboratory. The participant's AVPC-MS signature result (positive or negative) is required BEFORE randomization on to step 2. If sites receive a non-evaluable AVPC-MS signature result, SWOG Statistics and Data Management Center will provide instructions for resubmission * STEP 1 SCREENING REGISTRATION: NOTE: As a part of the Oncology Patient Enrollment Network (OPEN) registration process the treating institution's identity is provided in order to ensure that the current (within 365 days) date of institutional review board approval for this study has been entered in the system * STEP 1 SCREENING REGISTRATION: Participants must be informed of the investigational nature of this study and must sign and give informed consent in accordance with institutional and federal guidelines. Documentation of informed consent via remote consent is allowed * For participants with impaired decision-making capabilities, legally authorized representatives may sign and give informed consent on behalf of study participants in accordance with applicable federal, local, and Central Institutional Review Board (CIRB) regulations * STEP 2 RANDOMIZATION: NOTE: Participants must be registered to step 2 randomization within 70 days after registration to step 1. Participants must plan to start protocol therapy no more than 14 days after step 2 randomization * STEP 2 RANDOMIZATION: Participants must have castrate levels of testosterone with a baseline level \< 50ng/dL within 28 days prior to step 2 randomization * STEP 2 RANDOMIZATION: Participants must have evidence for metastatic prostate cancer by bone scan and/or CT/MRI (i.e., soft tissue, visceral, lymph node). Visceral and/or soft-tissue metastases must be ≥ 1.0 cm in diameter and lymph nodes must be \> 1.5 cm diameter in the short axis. Scans must be obtained within 28 days prior to randomization * NOTE: All disease must be assessed and documented on the baseline/pre-registration tumor assessment form * STEP 2 RANDOMZIATION: Participants must have progressive disease (PD) in the opinion of the treating investigator according to any of the following criteria * Progression in measurable disease (RECIST 1.1 criteria). Patient with measurable disease must have at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded). Each lesion must be at least 10 mm when measured by computed tomography (CT) \[CT scan thickness no greater than 5 mm\] or magnetic resonance imaging (MRI). Lymph nodes should be ≥ 15 mm in short axis. Previously irradiated lesions, primary prostate lesion and bone lesions will be considered non-measurable disease * Progression in bone as evidenced by: * Appearance of 2 or more new bone lesions on bone scan (BS). If equivocal, they must be confirmed by other imaging modalities (CT; MRI), and/or repeat BS \> 4 weeks later * Appearance of a new lytic lesion(s) and/or increasing size of an existing lesion by CT/MRI, since AVPC tumors may produce lytic bone lesions that are not detected on conventional bone scans * Rising prostate-specific antigen (PSA) defined (Prostate Cancer Working Group 2 \[PCWG2\]) as at least two consecutive rises in PSA to be docume