。
May I pay for the book by check?
Literal
Book [attributive-の] payment [object-を] check [with-で] pay-[may-てもよい]-[polite]-[question-か]?
~てもよい ('may do / it's okay to do') is the polite permission-asking form, combining ~て + も (even if) + よい (good). Often contracted to ~てもいい. The full ~てもよいですか is formal. 小切手 ('check, cheque') is a Sino-Japanese compound. で marks the instrument of payment.