Step 1

We have to find the matrix

$A=\left[\begin{array}{cc}a& -2\\ 1& 3\end{array}\right],B=\left[\begin{array}{cc}4& -5\\ b& 7\end{array}\right]$

${A}^{2}=\left[\begin{array}{cc}a& -2\\ 1& 3\end{array}\right]\left[\begin{array}{cc}a& -2\\ 1& 3\end{array}\right]$

$=\left[\begin{array}{cc}{a}^{2}-2& -2a-6\\ a+3& -2+9\end{array}\right]=\left[\begin{array}{cc}{a}^{2}-2& -2a-6\\ a+3& 7\end{array}\right]$

${A}^{2}-B=\left[\begin{array}{cc}{a}^{2}-2& -2a-6\\ a+3& 7\end{array}\right]-\left[\begin{array}{cc}4& -5\\ b& 7\end{array}\right]$

$=\left[\begin{array}{cc}{a}^{2}-6& -2a-1\\ a+3-b& 0\end{array}\right]$

${B}^{2}=\left[\begin{array}{cc}4& -5\\ b& 7\end{array}\right]\left[\begin{array}{cc}4& -5\\ b& 7\end{array}\right]$

$=\left[\begin{array}{cc}16-5b& -20-35\\ 4b+7b& -5b+49\end{array}\right]$

$A+{B}^{2}=\left[\begin{array}{cc}a+16-5b& -57\\ 11b+1& -5b+52\end{array}\right]$

$A+{B}^{2}=\left[\begin{array}{cc}a+16-5b& -57\\ 11b+1& -5b+52\end{array}\right]$