22.7. (GDR, 77). Find all polynomials $P(x)$ that satisfy the identity

$$
x P(x-1) \equiv (x-2) P(x), \quad x \in \mathbf{R}
$$
22.7. Substituting into the original identity the values $x=0$; 2, we get that the polynomial $P(x)$ has roots 0 and 1, and therefore, is divisible by the polynomial $x^{2}-x$. Further, substituting into the identity the expression $P(x)=$ $=\left(x^{2}-x\right) Q(x)$, we obtain for the polynomial $Q(x)$ the identity $Q(x) \equiv$ $=Q(x-1)$; from this we have $Q(0)=Q(-1)=Q(-2)=\ldots$ Therefore, $Q(x)=a$ - a constant, and the desired polynomials have the form $P(x)=$ $=a\left(x^{2}-x\right)$. (Verification shows that all polynomials of this form satisfy the required identity.)