1. Find all polynomials satisfying $(x-1) \cdot p(x+1)-(x+2) \cdot P(x) \equiv 0, x \in \mathbf{R}$.
1. Substituting $x=1,-2,0$ into the given identity, we can see that the polynomial has roots $0,1,-1$, meaning it is divisible by $x^{3}-x$. Next, substituting $p(x)=$ $\left(x^{3}-x\right) Q(x)$ into the identity, we can see that the polynomial $Q(x)$ satisfies the identity $Q(x) \equiv Q(x-1)$, from which we get $Q(0)=$ $Q(-1)=Q(-2)=\cdots$ Therefore, $Q(x)=a$, where $a$ is a constant. Thus, the polynomial we are looking for 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.