The quadratic ${\mathrm{x}}^2+\mathrm{ax}+\mathrm{b}+1=0$ has roots which are positive integers, then $\left(a^2+b^2\right)$ canbe equal to

${\mathrm{x}}^2+\mathrm{ax}+\mathrm{b}+1=0 ,$ has positive integral roots $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$

Now, $(\mathrm{\alpha }+\mathrm{\beta })=-\mathrm{a}\text{ and }\mathrm{\alpha \beta }=\mathrm{b}+1$

$\begin{array}{l}\Rightarrow (\mathrm{\alpha }+\mathrm{\beta }{)}^2+(\mathrm{\alpha \beta }-1{)}^2={\mathrm{a}}^2+{\mathrm{b}}^2\\ \Rightarrow {\mathrm{a}}^2+{\mathrm{b}}^2=\left({\mathrm{\alpha }}^2+1\right)\left({\mathrm{\beta }}^2+1\right)\end{array}$

$\Rightarrow {\mathrm{a}}^2+{\mathrm{b}}^2$ can be equal to 50 (since other options have prime number)