The values of a for which the matrix  $\mathrm{A}=\left(\begin{array}{ccc}\mathrm{a}& {\mathrm{a}}^2-1& -3\\ \mathrm{a}+1& 2& {\mathrm{a}}^2+4\\ -3& 4\mathrm{a}& -1\end{array}\right)$ is symmetric are 

${\mathrm{A}}^{\mathrm{\prime }}=\mathrm{A}$

$\begin{array}{l}\Rightarrow \left(\begin{array}{ccc}\mathrm{a}& \mathrm{a}+1& -3\\ {\mathrm{a}}^2-1& 2& 4\mathrm{a}\\ -3& {\mathrm{a}}^2+4& -1\end{array}\right)=\left(\begin{array}{ccc}\mathrm{a}& {\mathrm{a}}^2-1& -3\\ \mathrm{a}+1& 2& {\mathrm{a}}^2+4\\ -3& 4\mathrm{a}& -1\end{array}\right)\\ \Rightarrow \mathrm{a}+1={\mathrm{a}}^2-1\text{ and }4\mathrm{a}={\mathrm{a}}^2+4\\ \Rightarrow \mathrm{a}+1=4\mathrm{a}-4-1\Rightarrow 6=3\mathrm{a}\Rightarrow \mathrm{a}=2.\end{array}$