Let
![m \neq 0](/media/m/d/b/0/db0790fd1044c5a4b1e48c49fdd4d43c.png)
be an integer. Find all polynomials
![P(x)](/media/m/c/d/7/cd7664875343d44cd5f96a566b582b0e.png)
with real coefficients such that
![(x^3 - m x^2 + 1)P(x + 1) + (x^3 + m x^2 + 1)P(x - 1) = 2(x^3 - m x + 1)P(x)](/media/m/1/8/3/1835779a4c796a247069a55fc56a6f9b.png)
for all real numbers
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
.
%V0
Let $m \neq 0$ be an integer. Find all polynomials $P(x)$ with real coefficients such that $$
(x^3 - m x^2 + 1)P(x + 1) + (x^3 + m x^2 + 1)P(x - 1) = 2(x^3 - m x + 1)P(x)
$$ for all real numbers $x$.