Step 1:
Characterstic equation =$m^2+7m+12=0$
$\Rightarrow (m+3)(m+4)=0$
$m_1=-3,m_2=-4$
Therefore the CF=$Ae^{-3x}+Be^{-4x}$
Step 2:
The parameter integral is $\large\frac{1}{D^2+7D+12}$$e^{2x}$
$\alpha =2\neq m_1$ or $m_2$
$\therefore PI=\large\frac{1}{4+14+12}e^{2x}=\frac{e^{2x}}{30}$
Step 3:
G.S: $y=Ae^{-3x}+Be^{-4x}+\large\frac{e^{2x}}{30}$