= Solution
The <characteristic equation> has a repeated root $-1$, suggesting the substitution $y=e^{-x}v$. By the <product rule>,
$$
y'=e^{-x}(v'-v),\qquad y''=e^{-x}(v''-2v'+v),
$$
so $y''+2y'+y=e^{-x}v''$. The <inhomogeneous linear differential equation> reduces to $v''=2x$. Two integrations give
$$
v=\frac{x^3}{3}+C_1x+C_2.
$$
Therefore
$$
\boxed{y(x)=e^{-x}\left(\frac{x^3}{3}+C_1x+C_2\right)}.
$$
The two arbitrary constants account for the repeated-root <homogeneous solution>.
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