Okay, I have P and Q open sets in A such that A is contained in (P intersect Q), P intersect A is not empty, Q intersect A not empty, and P intersect Q is empty. I know that P' and Q' are relatively open and closed, (because P' is A intersect P with A closed and P open?) same for Q'. which would imply that A is disconnected because more than the whole set and empty set are relatively open and closed, but does it matter that the question says in a compact metric space A is a subset?
Though you will have to showed that they are also relatively closed. (Hint: P' and Q' are complement of each other in A).
As far as the part of A being compact, I am not so sure. I suspect that it is to make sure the set is not "too big" as you may have problems in some cases. It will have to leave it to someone who is more knowledgeable to answer. Sorry.