 Solution

For real solutions work on $x>-1$. Dividing by $2(1+x)$ puts the linear ordinary differential equation into the formIts integrating factor isThe product rule therefore givesTwice using integration by parts, or differentiating the proposed antiderivative, gives $\int x^2{e}^{-x}dx=-e^{-x}(x^2+2x+2)$. HenceThe endpoint $x=-1$ is singular. A finite continuous extension would require $C=e/2$, but then, with $h=x+1\downarrow 0$, $y=\frac{1}{2}\sqrt{h}+O(h^{3/2})$. Its derivative diverges, so even this choice has no differentiable extension to that endpoint.