Question $1$

Show that $y=K{e}^{-x}-2e^x+4x{e}^x$ is a solution of $\frac{dy}{dx}+y=8x{e}^x,$ and then find $K$ so that $y=K{e}^{-x}-2e^x+4x{e}^x$ also satisfies the boundary condition $y(1)=0.$