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Can I take out the books?
Literal
Books [topic-は] carrying-out can-do-[polite]-[question-か]?
持ち出し is the noun form of 持ち出す ('take out, carry away'). The pattern [noun]できる ('can do [noun]-suru verb') treats the noun as a する-verb base. Here 持ち出しできる = 'can take out' — common in library contexts when asking about borrowing or check-out privileges.