6. Find all non-zero polynomials $P(x)$ that satisfy $P\left(x^{2}-2 x\right) \equiv(P(x-2))^{2}, x \in \mathbf{R}$.
6. Let $y=x-1, Q(y)=P(y-1)$, then we have $(P(x-2))^{2}=(P(y-1))^{2}=(Q(y))^{2}$, $P\left(x^{2}-2 x\right)=P\left(y^{2}-1\right)=Q\left(y^{2}\right)$.

Thus, the original identity is transformed into $Q\left(y^{2}\right) \equiv(Q(y))^{2}, y \in \mathbf{R}$.
It is exactly the same as the identity in the previous problem. Therefore, $Q(y)=y^{n}$, hence $P(y)=(y+1)^{n}, n \in \mathbf{Z}^{+}$.