Solution
= Solution
Since $x>-1$, divide by $x+1$ to obtain the first-order <linear ordinary differential equation>
$$
y'-\frac3{x+1}y=(x+1)^4.
$$
An <integrating factor> is $\mu(x)=e^{-3\log(x+1)}=(x+1)^{-3}$. Multiplication by this factor gives
$$
\frac{d}{dx}\left[(x+1)^{-3}y\right]=x+1.
$$
After <integration>, $(x+1)^{-3}y=\tfrac12(x+1)^2+C$, so
$$
\boxed{y(x)=\frac12(x+1)^5+C(x+1)^3,\qquad C\in\mathbb R,\quad x>-1.}
$$