Trastuzumab, Ixabepilone, and Carboplatin in Treating Patients With HER2/Neu-Positive Metastatic Breast Cancer

Inclusion Criteria: * Patients with histologically confirmed adenocarcinoma of the breast which is metastatic and is known to overexpress HER2/neu who have received no prior chemotherapy for metastatic breast cancer; prior hormonal therapy for metastatic disease is allowed; NOTE: for this protocol, HER2 overexpression will be defined as 3+ HER2 positivity as measured by immunohistochemistry using the HercepTest (DAKO) or HER2 gene amplification as measured by fluorescent in-situ hybridization (FISH, e.g. Vysis); representative diagnostic tissue must be submitted for central diagnostic review * Patients must not be pregnant or breast feeding because of the teratogenic potential of these drugs; it is recommended that all females of childbearing potential have a blood test or urine study within 2 weeks prior to registration to rule out pregnancy; women of childbearing potential and sexually active males must be strongly advised to use an accepted and effective non-hormonal method of contraception * Patients must have at least one objective measurable disease parameter; baseline measurements and evaluations using RECIST criteria guidelines must be obtained within 4 weeks prior to registration to the study; NOTE: all areas of disease should be recorded and followed * Patients must have an ECOG performance status of 0 or 1 * Patients must be disease free of prior malignancy for \>= 5 years with the exception of curatively treated basal cell carcinoma or squamous cell carcinomas of the skin or carcinoma in situ of the cervix * Patients must not have a history of untreated brain metastasis or brain metastasis currently undergoing radiation; patients with brain metastasis representing the sole site of disease are not eligible for this study; patients with previously treated brain metastasis who have responded to brain radiotherapy and/or surgery and continue in response are eligible provided the brain is not the only site of measurable disease * Patients must not have peripheral neuropathy of any grade * Patients must not have a history of prior severe (grade 3 or 4) hypersensitivity reaction to a drug formulated in polyoxyethylated castor oil (Cremophor EL) * Patients must have left ventricular ejection fraction by MUGA scan or echocardiogram that is at or above the lower institutional limits of normal obtained within 6 weeks prior to registration * Patients must not have a history of New York Heart Association class 3 or 4 heart failure * Serum creatinine =\< 1.5 mg/dl * Granulocytes \>= 1500/mm\^3 * Platelets \>= 100,000/mm\^3 * SGOT(AST) and SGPT(ALT) =\< 1.5 x upper limit of normal (unless liver is involved by tumor, in which case SGOT(AST) and SGPT(ALT) can be =\< 2.0 x upper limit of normal) * Patients must have no history of prior therapy with trastuzumab (Herceptin), Ixabepilone (BMS-247550) or carboplatin for metastatic disease; patients who develop metastatic disease =\< 6 months after completing adjuvant trastuzumab (Herceptin), paclitaxel, docetaxel, carboplatin, or Ixabepilone (BMS-247550) are considered to have had prior therapy for metastatic disease and are excluded from study participation * Patients must not have received a cumulative dose of doxorubicin of greater than 360 mg/m\^2 or epirubicin of greater than 640 mg/m\^2 * Concurrent use of hormonal therapy is not permitted; concurrent radiation therapy is not permitted; hormonal therapy must have been discontinued \>= 1 week prior to registration; radiation therapy must have been completed \>= 2 weeks prior to registration * Patients may have had prior radiation therapy, but the previously irradiated tumors cannot be used to assess a clinical response; patients will not be eligible if they do not have other areas of measurable disease; an exception will be given for patients who have had tumor recurrence in an area that received adjuvant radiation treatments, such as the axilla or chest wall