$(10$ points$)$ This problem is related to Problems $8\mathrm{.}9-11$ in the text.

Consider the function F$($s$)=13$s$2+102$s$+229($s$+4)($s$2+12$s$+61)$F$($s$)=13$s$2+102$s$+229($s$+4)($s$2+12$s$+61).$

Find the partial fraction decomposition of F$($s$)$F$($s$).$ Enter all factors as first order terms in ss$,$ that is$,$ all terms should be of the form cs$$pcs$$p$,$ where cc is a constant and the root pp is a constant. Both cc and pp may be complex.

$13$s$2+102$s$+229$s$3+16$s$2+109$s$+244=13$s$2+102$s$+229$s$3+16$s$2+109$s$+244=++++$

Find the inverse Laplace transform of F$($s$)$F$($s$).($Remember to use u$($t$).$u$($t$).$

f$($t$)=$L$1\{$F$($s$)\}=$f$($t$)=$L$1\{$F$($s$)\}=$