Yes it is correct, She always loses her keys
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I would use
No, no me gusta la biblioteca.
I am hispanic and know spanish from an early age and this is the only one that makes sense.
P.S. In the question it is asking you if you like the library and 1 says Yes, I likes the library 2. Yes, you like the library 3. No I do not like the library and 4 No, you do not like the library, therefore number 3 is correct.
Sorry if that was a little to much
