Answer
$y(x)=C_1e^{-x}+C_2xe^{-x}+C_3\cos \sqrt 3x+C_4 \sin \sqrt 3x$
Work Step by Step
Solve the auxiliary equation for the differential equation. $$(r^2+3)^2(r+1)=0$$
Factor and solve for the roots. $$(r+\sqrt 3i)(r-\sqrt 3i)(r+1)^2$$
Roots are: $r_1=-1$, as a multiplicity of $2$ and $r_2=-\sqrt 3i, r_3=\sqrt 3 i$ as a multiplicity of $1$.
This implies that there are two independent solutions to the differential equation $y_1(x)=e^{-x}$ and $y_2=xe^{-x}$ and $y_3=\cos \sqrt 3x$ and $y_4= \sin \sqrt 3x$
Therefore, the general equation is equal to $y(x)=C_1e^{-x}+C_2xe^{-x}+C_3\cos \sqrt 3x+C_4 \sin \sqrt 3x$