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I hid my report card in the drawer so mother wouldn't see it.
Literal
Mother [subject-が] won't-see [so that-ように] I [topic-は] report-card [obj-を] drawer [in-に] put-away.
見ないように with the negative verb + ように expresses 'so that X doesn't happen' — the purposive ~ように reaching into the negative form. しまう here is the main verb ('put away, tuck away'), not the auxiliary ~てしまう; context and the direct object tell them apart. 通知票 is the school report card — culturally loaded item, often hidden from parents.