Neratinib and Paclitaxel With or Without Pertuzumab and Trastuzumab Before Combination Chemotherapy in Treating Patients With Metastatic or Locally Advanced Breast Cancer

Inclusion Criteria: * Histological confirmation of breast cancer * Able to provide written informed consent for the trial * Performance status of =\< 1 on the Eastern Cooperative Oncology Group (ECOG) performance scale * Able to swallow oral medication * Left ventricular ejection fraction (LVEF) assessment by 2-dimensional (D) echocardiogram or multi-gated acquisition (MUGA) scan performed within 90 days prior to registration must be \>= 50% * Absolute neutrophil count \>= 1,500 /uL * Platelets \>= 100,000 /uL * Hemoglobin \>= 9 g/dL * Creatinine clearance \>= 50 ml/min * Total bilirubin =\< 1.5 X upper limit of normal (ULN), for patients with congenital unconjugated hyperbilirubinemia (Crigler-Najjar syndrome type 1 and 2, Gilbert syndrome) that transient hyperbilirubinemia can occur due to physiological condition, as long as there is clear documentation of diagnosis, allowed to be enrolled if direct (conjugated) bilirubin is =\< 1.5 X ULN * Alanine aminotransferase and aspartate aminotransferase =\< 2.5 X ULN except in patients with aspartate aminotransferase (AST)/alanine aminotransferase (ALT) elevation that is declared to be caused due to liver metastasis, they are allowed to be enrolled as long as \< 5 x ULN * Subject of childbearing potential should be willing to use effective methods of birth control or be surgically sterile, or abstain from heterosexual activity for the course of the study and through at least 4 months after the last dose of study drug; subject of c…