Solve in general
$\displaystyle \frac{d^2y}{dx^2}+2\frac{dy}{dx}=1+e^{-2x}$
Solve the homogeneous equation first:
$\displaystyle y^{\prime\prime}+2y^{\prime}=0\implies r^2+2r=0\implies r=0$ or $\displaystyle r=-2$.
So we have $\displaystyle y_c=c_1+c_2e^{-2x}$.
Now, we solve the non-homogeneous equation by method of undetermined coefficients.
Take $\displaystyle y_p=Ax+Be^{-2x}+Cxe^{-2x}$, substitute it into the differential equation, and then solve for the unknown coefficients.
Your final answer will be of the form $\displaystyle y=y_c+y_p=\dots$
Can you take it from here?