Think the hint that the guest gave helps tremendously, but, I digress.
P(x) = x^2. Then P(1) = 1, p(2) = 4, p(3) = 9, which makes p(n) always a perfect square. This works for all polynomials that factor, for example P(x) = x^3.
PS: I did some more googling and found out that this question was posted IN OCTOBER OF LAST YEAR!!!!!