Given $t_n=n^2+2^n$.
$\therefore\:$ Sum of $n$ terms of the series $=S_n=\sum t_n=\sum (n^2+2^n)$
We know that $\sum (A+B)=\sum A+\sum B$
$=\sum n^2+\sum 2^n$
$\sum n^2=1^2+2^2+3^2+..........=\large\frac{n(n+1)(2n+1)}{6}$
$\sum 2^n=2+2^2+2^3+2^4+..........$ which is a G.P. with common ratio $2$
$\qquad=\large\frac{2.(2^n-1)}{2-1}$$=2.(2^n-1)$
$\Rightarrow\:S_n=\large\frac{n(n+1)(2n+1)}{6}$$+2.(2^n-1)$