Find all polynomials $P$ such that

$$
P\left(x^{2}-2 x\right)=(P(x-2))^{2}
$$
A $Q(x)=P(x-1)$ polynomial (1) implies the condition $Q\left(x^{2}-2 x+1\right)=(Q(x-1))^{2}$, which is equivalent to

$$
Q\left(x^{2}\right)=(Q(x))^{2}
$$

condition. Let

$$
Q(x)=\sum_{i=1}^{n} a_{i} x^{i}
$$

where $a_{n} \neq 0$, then according to (2)

$$
\sum_{i=1}^{n} a_{i} x^{2 i}=\left(\sum_{i=1}^{n} a_{i} x^{i}\right)^{2}
$$

The coefficient of the $x^{2 n}$ term on the left side is $a_{n}$, and on the right side it is $a_{n}^{2}$, which only holds for $a_{n}=1$ given $a_{n} \neq 0$. We will show that all other coefficients are 0. If, after $a_{n}$, $a_{k}$ were the first non-zero coefficient, then the coefficient of $x^{n+k}$ on the right side would be non-zero, while on the left side it would be 0.

Therefore, the solutions to (2) are the polynomials $Q(x)=x^{n}$, and the solutions to (1) are the polynomials $P(x)=(x+1)^{n}$, for any exponent $n$.