Given equation, $x^{3}-x^{2}-x-2=0$
$\Rightarrow (x-2)\left(x^{2}+x+1\right)=0$
$\therefore \alpha$ and $\beta$ are $\frac{-1 \pm \sqrt{3} i}{2}$ or we can say
$\alpha$ and $\beta$ are non-real complex roots of unity.
So, let $\alpha=\omega$ and $\beta=\omega^{2}$,
where $\omega^{3}=1$ and $\omega^{2}+\omega+1=0$
$\therefore \alpha^{2020}+\beta^{2020}+\alpha^{2020} \beta^{2020}$
$=\omega^{2020}+\omega^{4040}+\omega^{2020} \omega^{4040}$
$=\left(\omega^{3}\right)^{673} \omega+\left(\omega^{3}\right)^{1346} \omega^{2}+\left(\omega^{3}\right)^{673} \omega\left(\omega^{3}\right)^{1346} \omega^{2}$
$=\omega+\omega^{2}+\omega^{3}=1+\omega+\omega^{2}=1+\alpha+\beta$