VTRMC 1980 Problem 4

Let $P(x)$ be any polynomial of degree at most $3.$ It can be shown that there are numbers $x_1$ and $x_2$ such that $\int_{-1}^1 P(x) \ dx = P(x_1) + P(x_2),$ where $x_1$ and $x_2$ are independent of the polynomial $P.$ (a) Show that $x_1=-x_2.$ (b) Find $x_1$ and $x_2.$