$\alpha$ and $\beta$ are the roots of the equation $x^2-ax+b=0$. If ${\alpha }^2+{\beta }^2$ and ${\alpha }^3+{\beta }^3$ are the roots of the equation ${\mathrm{Ax}}^2+\mathrm{Bx}+\mathrm{C}=0$, then C =

$\alpha$ and $\beta$ are the roots of the equation $x^2-ax+b=0$

$\alpha +\beta =a$ and $\alpha \beta =b$

${\alpha }^2+{\beta }^2={\left(\alpha +\beta \right)}^2-2\alpha \beta$

$=a^2-2b$

${\alpha }^3+{\beta }^3={\left(\alpha +\beta \right)}^3-3\alpha \beta \left(\alpha +\beta \right)$

$=a^3-3ab$

Equation whose roots are  ${\alpha }^2+{\beta }^2$ and ${\alpha }^3+{\beta }^3$  then

$x^2-x\left({\alpha }^2+{\beta }^2+{\alpha }^3+{\beta }^3\right)+\left({\alpha }^2+{\beta }^2\right)\left({\alpha }^3+{\beta }^3\right)=0$

$x^2-x\left(a^2-2b+a^3-3ab\right)+\left(a^2-2b\right)\left(a^3-3ab\right)=0$