This randomized phase III trial studies docetaxel, carboplatin, trastuzumab, and pertuzumab with estrogen deprivation to see how they work compared to docetaxel, carboplatin, trastuzumab, and pertuzumab without estrogen deprivation in treating patients with hormone receptor-positive, human epidermal growth factor receptor 2 (HER2)-positive breast cancer that is operable or has spread from where it started to nearby tissue or lymph nodes (locally advanced). Drugs used in chemotherapy, such as docetaxel, carboplatin, trastuzumab, and pertuzumab, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Giving more than one drug (combination chemotherapy) may kill more tumor cells. Estrogen can cause the growth of breast cancer cells. Hormone therapy using goserelin acetate and aromatase inhibition therapy may fight breast cancer by blocking the use of estrogen by the tumor cells. Radiation therapy uses high energy x rays to kill tumor cells. Giving combination chemotherapy and radiation therapy with or without hormone therapy may be an effective treatment for hormone receptor-positive, HER2-positive, operable or locally advanced breast cancer.