Question 73
Let $m\ne0$ be an integer. Find all polynomials $P(x)$ with real coefficients such that
$$\begin{aligned}
&(x^3-mx^2+1)P(x+1)\\
&\quad +(x^3+mx^2+1)P(x-1)\\
&=2(x^3-mx+1)P(x)
\end{aligned}$$
for all real numbers $x$.
$$\begin{aligned}
&(x^3-mx^2+1)P(x+1)\\
&\quad +(x^3+mx^2+1)P(x-1)\\
&=2(x^3-mx+1)P(x)
\end{aligned}$$
for all real numbers $x$.