5・142 Find all non-zero polynomials $P(x)$ that satisfy
$$P\left(x^{2}-2 x\right) \equiv(P(x-2))^{2}, x \in R$$
[Solution] Let $y=x-1, Q(y)=P(y-1)$, then we have
$$\begin{array}{l}
\left(P(x-2)^{2}\right)=(P(y-1))^{2}=(Q(y))^{2} \\
P\left(x^{2}-2 x\right)=P\left(y^{2}-1\right)=Q\left(y^{2}\right)
\end{array}$$

Thus, the original identity is transformed into
$$Q\left(y^{2}\right) \equiv(Q(y))^{2}, y \in R$$

From the previous problem, we know that $Q(y)=y^{n}$.
Therefore, $P(y)=(y+1)^{n}, n \in Z^{+}$.
Hence, $P(x)=(x+1)^{n}, n \in Z^{+}$.