BxMO 2010 Problem 2

Find all polynomials $p(x)$ with real coefficients such that

$$p(a+b-2 c)+p(b+c-2 a)+p(c+a-2 b)=3 p(a-b)+3 p(b-c)+3 p(c-a)$$

for all $a, b, c \in \mathbb{R}$.