Homogeneous solution :
Characteristic polynomial –
$r^{2}-4r+4=0 \\ =>r=1, 3$
$a_{n}^{h} = \alpha +\beta .3^{n}$
Particular Solution :
$a_{n}^{p} = c.2^{n}+n(dn+e)$ [Multiplying $dn+e$ by $n$ as 1 is already root of the char. polynomial]
After putting $a_{n}^{p}$ in our recurrence relation and solving, we get
$c=-4 \hspace{10mm}d=-1/4\hspace{10mm}e=-5/2$
Therefore,
$a_{n}^{p} = -4.2^{n}-n^{2}/4-5n/2$
Complete solution:
$a_{n} = a_{n}^{p}+a_{n}^{h} = -4.2^{n}-n^{2}/4-5n/2 + \alpha +\beta .3^{n}$
Substituting the base conditions, we get $\alpha=1/8 $ and $\beta=39/8$
$\therefore a_{n} =4.2^{n}-\frac{n^{2}}{4}-\frac{5n}{2} + \frac{1}{8} +\frac{39}{8}3^{n}$