Example 8 Find all non-zero polynomials $P(x)$ that satisfy $P\left(x^{2}-2 x\right) \equiv[P(x-2)]^{2}(x \in \mathbf{R})$.
(1976 Bulgarian Competition Problem)
To solve this example, let $y=x-1$, then $Q(y)=P(y-1), [P(x-2)]^{2}=[P(y-1)]^{2}=[Q(y)]^{2}, P\left(x^{2}-2 x\right)=P\left(y^{2}-1\right)=Q\left(y^{2}\right)$, transforming it into the form of Example 5, thus $P(x)=(x+1)^{n}$ is the solution sought.