First we find the eigen values of A

To find eigen values of A ,characteristic eqation is , |A- λI| =0 , where λ is the eigen value of A & I is the identity matrix

i.e. $$ \begin{vmatrix} 1-λ&0&0 \\ 2&-2-λ&0 \\ 3&5&3-λ \end{vmatrix} = 0 $$ Expanding we get a cubic equation in λ as $ λ^3 - 2λ^2 - 5λ + 6 =0$ i.e. λ = 1, -2, 3

If λ is the eigen value of A ,then by the property eigen values of $A^2 +2I$ will be $λ^2 +2$

By putting λ = 1, -2, 3 , Eigen values of $A^2 +2I$ are 3, 6, 11