2013 IMO Shortlist A6

Let $m \neq 0$ be an integer. Find all polynomials $P(x)$ with real coefficients such that $$\left(x^{3}-m x^{2}+1\right) P(x+1)+\left(x^{3}+m x^{2}+1\right) P(x-1)=2\left(x^{3}-m x+1\right) P(x)$$ for all real numbers $x$.

设 $m \neq 0$ 为整数。查找所有实数系数满足 $$\left(x^{3}-m x^{2}+1\right) P(x+1)+\left(x^{3}+m x^{2}+1\right) P(x-1)=2\left(x^{3}-m x+1\right) P(x)$$ 对于所有实数 $x$ 的多项式 $P(x)$。