If $\alpha$ and $\beta$ are the real roots of the equation $x^2-(k-2)x+(k^2+3k+5)=0(k\in R)$ then maximum value of ${\alpha }^2+{\beta }^2$ is

$\begin{array}{l}\alpha +\beta =k-2,\alpha \beta =k^2+3k+5\\ {\alpha }^2+{\beta }^2={(\alpha +\beta )}^2-2\alpha \beta ={(k-2)}^2-2k^2-6k-10\\ =-k^2-10k-6\\ Letf\left(k\right)=-k^2-10k-6\\ f^1\left(k\right)=0\Rightarrow -2k-10=0\Rightarrow k=-5\\ f^{11}\left(k\right)=-2<0\end{array}$

Maximum value $\begin{array}{l}f(-5)=-25+50-6\\ =19\end{array}$