This randomized phase III trial studies doxorubicin hydrochloride, cyclophosphamide, paclitaxel, and trastuzumab to see how well they work compared to combination chemotherapy alone in treating women with breast cancer that is human epidermal growth factor receptor 2 (HER2)-positive and has spread to the lymph nodes or high-risk and has not spread to the lymph nodes. Drugs used in chemotherapy use different ways to stop tumor cells from dividing so they stop growing or die. Monoclonal antibodies such as trastuzumab can locate tumor cells and either kill them or deliver tumor-killing substances to them without harming normal cells. It is not yet known whether combination chemotherapy is more effective with or without trastuzumab in treating breast cancer.