The correct option is
A everywhere
The given sentence indicates that the subject has been looking for the object. As we can see, the answer will modify the verb 'looking'. Thus, the answer must be an adverb. Also, the sentence suggests that the subject was searching for the object in certain places. Thus, the answer must be an adverb of place.
Option A is correct. 'everywhere' is an adverb of place. It suggests that the subject looked for the object at every place he/she could think of. Thus, it conveys the correct meaning and is the correct answer.
Option B is incorrect. 'all pieces' is not an adverb. 'all' is an adjective of quantity and 'pieces' is a noun. Thus, it cannot be the answer.
Option C is incorrect. 'anywhere' is an adverb of place but, it indicates that the subject searched for the object at any random places. This does not make much sense. Also, anywhere is generally used in negative sentences. For these reasons, it is incorrect.
Option D is incorrect. 'some places' is not an adverb. 'some' is an adjective of quantity and 'places' is a noun. Thus, it cannot be the answer.