\(\Delta'=\left(m-3\right)^2-\left(m^2-2m+2\right)=-4m+7\ge0\Rightarrow m\le\frac{7}{4}\)
\(\left\{{}\begin{matrix}x_1+x_2=2\left(m-3\right)\\x_1x_2=m^2-2m+2\end{matrix}\right.\)
\(A=x_1^2+x_2^2=\left(x_1+x_2\right)^2-2x_1x_2\)
\(=4\left(m-3\right)^2-2\left(m^2-2m+2\right)\)
\(=2m^2-20m+32=2\left(m-\frac{33}{4}\right)\left(m-\frac{7}{4}\right)+\frac{25}{8}\)
Do \(m\le\frac{7}{4}\Rightarrow\left\{{}\begin{matrix}m-\frac{33}{4}< 0\\m-\frac{7}{4}\le0\end{matrix}\right.\) \(\Rightarrow2\left(m-\frac{33}{4}\right)\left(m-\frac{7}{4}\right)\ge0\)
\(\Rightarrow A\ge\frac{25}{8}\Rightarrow A_{min}=\frac{25}{8}\) khi \(m=\frac{7}{4}\)
