22.8. (New York, 76). Find all polynomials $P(x)$ satisfying the identity

$$
(x-1) P(x+1)-(x+2) P(x) \equiv 0, \quad x \in \mathbf{R}
$$

3 Edited by I. N. Sergeev
22.8. Substituting into the original identity the values $x=1; -2; 0$, we get that the desired polynomial $P(x)$ has roots 0, $\pm 1$, which means it is divisible by the polynomial $x^{3}-x$. Next, substituting into the identity the expression

$$
P(x)=\left(x^{3}-x\right) Q(x),
$$

we obtain for the polynomial $Q(x)$ the identity $Q(x+1)-Q(x)=0$, from which we have $Q(0)=Q(1)=Q(2)=\ldots$ Therefore, $Q(x)=a$ - a constant, and the desired polynomials have the form $P(x)=a\left(x^{3}-x\right)$. (Verification shows that all polynomials of this form satisfy the required identity.)