Ganetespib, Paclitaxel, Trastuzumab and Pertuzumab for Metastatic Human Epidermal Growth Factor Receptor 2 Positive Breast Cancer

Inclusion Criteria:

* Pathologically confirmed diagnosis of breast cancer (central confirmation is not required)
* Patients must be at least 18 years of age
* Metastatic or advanced breast cancer that is evaluable OR metastatic or advanced breast cancer that is measurable for response as per Response Evaluation Criteria in Solid Tumors (RECIST) 1.1
* Eastern Cooperative Oncology Group (ECOG) performance status 0-2
* Life expectancy of at least 3 months as assessed by the investigator
* Patients with estrogen receptor (ER)+ breast cancer must have received prior treatment with at least one hormone therapy
* Absolute neutrophil count ≥ 1,500 cells/uL
* Platelets ≥ 100,000/uL
* Hemoglobin ≥ 9.0g/dL
* Total bilirubin ≤ 1.5 x the upper limit of normal (ULN)
* Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) ≤ 2.5 x ULN
* Albumin ≥ 3.0 g/dL
* Adequate renal function as defined by a serum creatinine ≤ 1.5 x ULN
* Willingness and ability to comply with scheduled visits, treatment plans, laboratory tests, and other study procedures
* Female subjects of childbearing potential and males must agree to use adequate contraception (e.g., hormonal or barrier method of birth control; abstinence) for the duration of study treatment
* Female subjects of childbearing age must have a negative serum pregnancy test at study entry
* Patients with human immunodeficiency virus (HIV)/acquired immune deficiency syndrome (AIDS) are allowed on study if they have an undetectable viral load, cluster of differentiation (CD)4 \> 300 and are on a stable highly active antiretroviral therapy (HAART) regimen for 1 month prior to study enrollment
* Patients are required to have HER2+ breast cancer defined as a fluorescent in situ hybridization (FISH)- ratio of \>= 2.0 or immunohistochemistry (IHC) 3+
* Patients for the triplet regimen (ganetespib, paclitaxel, trastuzumab):

  * Any number of prior chemotherapies or biological therapies are allowed, patients are required to have prior treatment with pertuzumab and ado-trastuzumab emtansine with the exceptions listed below:

    * Metastatic patients who have not received prior pertuzumab are eligible if: heavily pretreated prior to Food and Drug Administration (FDA) approval of pertuzumab (08-Jun-2012) for first line treatment of HER2+ MBC
    * Metastatic patients who have not received ado-trastuzumab emtansine are eligible if: heavily pretreated prior to FDA approval of ado-trastuzumab emtansine (22-Feb-2013) for the treatment of patients with HER2+ MBC who previously received trastuzumab and a taxane, separately or in combination
* Patients for the triplet regimen + pertuzumab:

  * Prior hormonal therapy for ER+ and/or PR+ HER2+ disease in the metastatic setting is allowed.
  * Prior T-DM1 in the metastatic setting is allowed if patients have progressed within 6 months after treatment for early-stage disease.
* No prior history of intolerance or hypersensitivity to trastuzumab and/or adverse events related to trastuzumab, murine proteins, or any of the excipients that resulted in trastuzumab being permanently discontinued

Exclusion Criteria:

* Fewer than 21 days since last anti-tumor therapy, including chemotherapy, biologic except trastuzumab, experimental, immune, radiotherapy for the treatment of breast cancer, with the following exceptions:

  * Hormone therapy
  * Palliative radiation therapy involving =\< 25% of marrow-bearing bone is allowed if completed within \>= 14 days prior to first study treatment
* Surgery, radiotherapy, or lesion ablative procedure to the only area of measurable/evaluable disease
* Major surgery within 4 weeks prior to first dose of ganetespib
* Poor venous access for study drug administration

  * Study drug administration via indwelling catheters is allowed only if the catheter is made of silicone material
* No prior chemotherapy in the metastatic setting is allowed.
* Prior pertuzumab is not allowed in the metastatic setting. Pertuzumab given in the neoadjuvant and/or adjuvant setting is allowed.
* History of intolerance or hypersensitivity to trastuzumab and/or pertuzumab
* Adverse events related to trastuzumab, murine proteins, or any of the excipients that resulted in trastuzumab being permanently discontinued
* History of severe (grade 3 or 4) allergic or hypersensitivity reactions to excipients (e.g., polyethylene glycol \[PEG\] 300 and polysorbate 80)
* History of intolerance or hypersensitivity to paclitaxel and/or adverse events related to paclitaxel that resulted in paclitaxel being permanently discontinued
* Peripheral neuropathy of grade \>= 2 per National Cancer Institute (NCI) Common Terminology Criteria for Adverse Events (CTCAE), version 4.0, at the time of or within 3 weeks prior to the first study therapy
* Baseline QTc \> 470 msec (average of triplicate ECG recordings); a consistent method of QTc calculation must be used for each patient's QTc measurements. QTcF (Fridericia's formula) is preferred
* Use of medications that have been linked to the occurrence of torsades de pointes

  * Patients will be eligible for the study if they discontinue any of the listed medications two weeks prior to registration and study enrollment
  * Stable regimen of antidepressants of the SSRI class is allowed (common SSRIs include escitalopram oxalate, citalopram, fluvoxamine, paroxetine, sertraline and fluoxetine)
* Left ventricular ejection fraction (EF) \< 50% at baseline
* Serum potassium, magnesium, and calcium levels (corrected for albumin) outside the laboratory's reference range despite correction
* Treatment with chronic immunosuppressants (e.g., cyclosporine following transplantation)
* Women who are pregnant or lactating
* Current known active infection with HIV, hepatitis B or C viruses
* Uncontrolled systemic disease (e.g., clinically significant cardiac, pulmonary or metabolic disease)
* Other medications, or severe acute/chronic medical or psychiatric condition, or laboratory abnormality that may increase the risk associated with study participation or study drug administration, or may interfere with the interpretation of study results in the judgment of the investigator
* History of clinically significant cardiac dysfunction, including:

  * Unstable angina
  * Unstable atrial fibrillation
  * Symptomatic bradycardia
  * Indwelling temporary pacemaker
  * History of MI within 6 months prior to first study treatment
  * History of symptomatic CHF (grade \> 3 by NCI CTCAE or Class \> II by New York Heart Association (NYHA) criteria
  * Ventricular tachycardia or a supraventricular tachycardia that requires treatment with a Class 1a antiarrhythmic drug (eg quinidine, procainamide, disopyramide) or Class III antiarrhythmic drug (eg sotalol, amiodarone, dofetilide). Use of other antiarrhythmic drugs is permitted
  * Second or third degree atrioventricular (AV) block unless treated with a permanent pacemaker
  * Complete left bundle branch block (LBBB)
  * History of long QT syndrome or a family member with this condition
* Brain metastases that are:

  * Progressive or
  * Have required any type of therapy (including radiation, surgery or steroids) to control symptoms from brain metastases within 60 days prior to the first study treatment
* History of an invasive second primary malignancy diagnosed within the previous 3 years, except for appropriately treated stage I endometrial or cervical carcinoma or prostate carcinoma treated surgically, and non-melanoma skin cancer