If $\alpha ,\beta$ are the roots of $x^2-2x+4=0$, then ${\alpha }^5+{\beta }^5=$

$\alpha ,\beta$ are roots of $a{x}^2+bx+c=0$

$\alpha +\beta =2,\alpha \beta =4$

$({\alpha }^3+{\beta }^3)({\alpha }^2+{\beta }^2)={\alpha }^5+{\alpha }^3{\beta }^2+{\beta }^2{\alpha }^3+{\beta }^5$

${\alpha }^5+{\beta }^5=({\alpha }^3+{\beta }^3)({\alpha }^2+{\beta }^2)-{\alpha }^2{\beta }^2(\alpha +\beta )\cdots \cdots \left(1\right)$

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

$=4-8=-4$

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

$=8-12\left(2\right)=8-24=-16$

${\alpha }^5+{\beta }^5=(-16)(-4)-16\left(2\right)$

$=64-32$

$=32$