Are you sure you have copied the problem correctly? The question of whether (A+B)(A-B)= A^2- B^2 would be much more interesting.
For either problem, the first thing you should have thought about was expanding the product: (A+B)(A-B)= A(A+B)+ B(A-B)= A(A)+ A(-B)+ BA+ B(-B)= A^2- AB+ BA- B^2. In order for that to be "A^2+ B^2", it must be true that -AB+ BA= 2B^2. Do you think it is at all likely that that will be true for all A? In order for that to be "A^2- B^2", It must be true that -AB+ AB= 0 which is the same as saying that AB= BA. Is that true for matrices?