If $\alpha and \beta$are non-real roots of ${\mathrm{x}}^3-{\mathrm{x}}^2-\mathrm{x}-2=0$ then  ${\mathrm{\alpha }}^{2020}+{\mathrm{\beta }}^{2020}+{\mathrm{\alpha }}^{2020}\cdot {\mathrm{\beta }}^{2020}=$

${\mathrm{x}}^3-{\mathrm{x}}^2-\mathrm{x}-2=0\Rightarrow (\mathrm{x}-2)\left({\mathrm{x}}^2+\mathrm{x}+1\right)=0$
$\begin{array}{l}\therefore \mathrm{\alpha }\text{ and }\mathrm{\beta }\text{ are }\frac{-1\pm \mathrm{i}\sqrt{3}}{2}\Rightarrow \mathrm{\alpha }=\mathrm{\omega },\mathrm{\beta }={\mathrm{\omega }}^2\\ {\mathrm{\alpha }}^{2020}+{\mathrm{\beta }}^{2020}+{\mathrm{\alpha }}^{2020}\cdot {\mathrm{\beta }}^{2020}={\left({\mathrm{\omega }}^3\right)}^{673}\cdot \mathrm{\omega }+{\left({\mathrm{\omega }}^3\right)}^{1346}\cdot {\mathrm{\omega }}^2+{\left({\mathrm{\omega }}^3\right)}^{2020}\\ =\mathrm{\omega }+{\mathrm{\omega }}^2+1=1+\mathrm{\alpha }+\mathrm{\beta }\end{array}$