Let $P(x)$ be a nonconstant polynomial, where all the coefficients are nonnegative integers. Prove that there exist infinitely many positive integers $n$ such that $P(n)$ is composite.

Let P(x) = x^2. Then P(1) = 1, p(2) = 4, p(3) = 9, so p(n) is always a perfect square. You can do this for any polynomial that factors, like P(x) = x^3 and P(x) = x(x + 1).