Twenty percent of women with breast cancer have triple negative breast cancer. The standard treatment for triple negative breast cancer is typically a combination of chemotherapy, surgery, +/- radiation therapy. When treated with standard therapy women with triple negative breast cancer have a worse long term outcomes as compared to women who do not have triple negative breast cancer. Triple negative breast cancer cells usually have a surface marker called EGFR (epidermal growth factor receptor). Women whose breast cancer cells have the EGFR surface marker have worse long term outcomes as compared to women whose tumors do not have the EGFR marker. Erlotinib (Tarceva) targets EGFR and is currently used for treatment of other cancers like lung and pancreas. This study will assess a combination of chemotherapy with erlotinib in women with triple negative breast cancer.