22.10. (NBR, 76). Find all non-zero polynomials $P(x)$ that satisfy the identity

$$
P\left(x^{2}-2 x\right) \equiv(P(x-2))^{2}, \quad x \in \mathbf{R}
$$
22.10. Let $y=x-1, Q(y)=P(y-1)$. Then we have

$$
\begin{aligned}
& (P(x-2))^{2}=(P(y-1))^{2}=(Q(y))^{2}, \\
& P\left(x^{2}-2 x\right)=P\left(y^{2}-1\right)=Q\left(y^{2}\right),
\end{aligned}
$$

and the original identity can be written as

$$
Q\left(y^{2}\right)=(Q(y))^{2}, \quad y \in \mathbb{R},
$$

which coincides, up to notation, with the identity of problem 22.9. Therefore, $Q(y)=y^{n}$ and $P(y)=(y+1)^{n}$, where $n \in \mathbb{Z}^{+}$.