5・138 Find all polynomials $P(x)$ that satisfy
$$x P(x-1) \equiv (x-2) P(x), x \in R$$
[Solution] Substituting $x=0,2$ into the given identity, we can see that the polynomial $P(x)$ has roots 0 and 1, meaning it is divisible by the polynomial $x^{2}-x$. Next, substituting $P(x) \equiv\left(x^{2}-x\right) \boldsymbol{Q}(x)$ into the identity, we get
$$\begin{array}{l}
x\left[(x-1)^{2}-(x-1)\right] \boldsymbol{Q}(x-1) \equiv(x-2)\left(x^{2}-x\right) \boldsymbol{Q}(x) \\
x(x-1)(x-2) \boldsymbol{Q}(x-1) \equiv x(x-1)(x-2) \boldsymbol{Q}(x)
\end{array}$$

Thus, the polynomial $Q(x)$ satisfies the identity
$$Q(x) \equiv Q(x-1)$$

This implies $Q(0)=Q(1)=Q(2)=\cdots$
Therefore, $Q(x) \equiv a$, where $a$ is a constant, and the desired polynomial is
$P(x)=a\left(x^{2}-x\right)$. Conversely, it is easy to verify that all such polynomials also satisfy the given identity.