5-139 Find all polynomials \( P(x) \) that satisfy
$$
(x-1) P(x+1)-(x+2) P(x) \equiv 0, x \in R
$$
[Solution] Substituting $x=1, -2, 0$ into the given identity, we can see that the polynomial has roots 0, $\pm 1$. That is, it is divisible by $x^{3}-x$. Next, substituting $P(x)=\left(x^{3}-x\right) Q(x)$ into the identity, we find that the polynomial $Q(x)$ satisfies the identity $Q(x) \equiv Q(x-1)$, which leads to
$$
Q(0)=Q(-1)=Q(-2)=\cdots
$$

Thus, $Q(x) \equiv a$, where $a$ is a constant, and hence, the polynomial we seek is $P(x)=a\left(x^{3}-x\right)$. Conversely, it is easy to verify that all polynomials of this form also satisfy the given identity.