polynomials - A contest math problem

Let $P(x)$ be a polynomial with integer coefficients of degree $d>0$.

1. If $\alpha $ and $\beta $ are two integers such that $P(\alpha)=1$ and $P(\beta)=-1$, then prove that $|\beta -\alpha | $ divides $2$.



2. Prove that the number of distinct integer roots of $(P(x))^2-1$ is at most $d+2$.

First one is very easy. But I cannot understand how to prove the second one. I would appreciate any help.