The key is to know which receptors each kinase inhibitor covers. HER1 is the epidermal growth factor receptor and HER2 is the ErbB2 receptor. We want one drug that silences both.
Lapatinib is the textbook dual inhibitor. It simultaneously blocks the kinase domains of EGFR (HER1) and HER2, which is why it works in HER2-positive breast cancer and is often combined with capecitabine. That fits the question exactly.
Erlotinib and gefitinib are narrower in scope; each clamps onto the EGFR (HER1) kinase alone and shines in lung cancer, but neither touches HER2. Canertinib is indeed a broad pan-ErbB blocker on paper, yet it stayed an experimental compound and is not an established clinical drug, so it is not the answer the examiner intends among the recognised agents.
$\therefore$ Lapatinib is the dual HER1 and HER2 inhibitor.
\[\boxed{\text{Lapatinib}}\]