Docetaxel, Carboplatin, Trastuzumab, and Lapatinib in Treating Patients With Early Stage Breast Cancer

DISEASE CHARACTERISTICS: * Histologically confirmed primary invasive adenocarcinoma of the breast fulfilling the following criteria: * Nonmetastatic disease * Operable and adequately excised * Patients with nonresectable deep margin invasion are eligible provided they have had or will receive radiotherapy to the region * Patients with histologically documented infiltration of the skin (pT4) are eligible provided they have undergone or will receive radiotherapy encompassing the tumor bed * Node-positive OR -negative and determined eligible to receive adjuvant trastuzumab (Herceptin®) * No positive or suspicious internal mammary nodes by SNS that have not been or will not be irradiated * No supraclavicular lymph node involvement (confirmed by fine needle aspiration or biopsy) * Over expression and/or amplification of HER2 in the invasive component of the primary tumor, according to one of the following: * 3+ over-expression by IHC (\> 30% of invasive tumor cells) * 2+ or 3+ (in 30% or less neoplastic cells) over-expression by IHC AND in situ hybridization (FISH/CISH) test demonstrating HER2 gene amplification * HER2 gene amplification by FISH/CISH (\> 6 HER2 gene copies per nucleus, or a FISH ratio \[HER2 gene copies to chromosome 17 signals\] of \> 2.2.) * Negative or equivocal overall result (FISH test ratio of \< 2.2, \< 6.0 HER2-gene copies per nucleus) and staining scores of 0, 1+, 2+, or 3+ (in 30% or less neoplastic cells) by IHC not allowed * Hormone receptor status known (estrogen receptor with or without progesterone receptor) PATIENT CHARACTERISTICS: * Menopausal status not specified * ECOG performance status 0-1 * Hemoglobin ≥ 10.0 g/dL * ANC ≥ 1,500/mm\^3 * Platelet count ≥ 100,000/mm\^3 * Serum creatinine ≤ 2.0 times upper limit of normal (ULN) * AST and ALT ≤ 2.5 times ULN * Alkaline phosphatase ≤ 2.5 times ULN * Bilirubin ≤ 1.5 times ULN (≤ 2.0 times ULN if known Gilbert syndrome) * Baseline LVEF ≥ 50% measured by ECHO or MUGA scan * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective contraception * No serious cardiac illness or medical condition including, but not limited to, any of the following: * History of documented congestive heart failure (any NYHA class) or systolic dysfunction (LVEF \< 50%) * High-risk uncontrolled arrhythmias (e.g., ventricular tachycardia, high-grade atrioventricular-block \[second degree or higher\], or supraventricular arrhythmias that are not adequately rate-controlled) * Angina pectoris requiring antianginal medication * Clinically significant valvular heart disease * Evidence of transmural infarction on ECG * Poorly controlled hypertension (any reading of systolic BP \> 180 mm Hg or diastolic BP \> 100 mm Hg) * No other concurrent serious diseases that may interfere with planned treatment including severe pulmonary conditions or illness * None of the following: * Ulcerative colitis * Malabsorption syndrome * Any disease significantly affecting gastrointestinal function * Inability to swallow oral medication PRIOR CONCURRENT THERAPY: * See Disease Characteristics * No prior mediastinal irradiation except internal mammary-node irradiation for the present breast cancer * No prior anti-HER2 therapy for any reason * No prior biologic or immunotherapy for breast cancer * No prior resection of the stomach or small bowel * No other concurrent anticancer therapy including chemotherapeutic agents, biologic agents, or radiotherapy * No concurrent anticancer treatment in another investigational trial with hormone therapy or immunotherapy unless approved by the study chair * No concurrent CYP3A4 inhibitors or inducers * No concurrent epoetin alfa, including darbepoetin alfa * No concurrent oprelvekin