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

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