If α, β, γ, are the roots of the equation x ^ 4 + x ^ 3 + x ^ 2 + x...

We know that ${x}^{4} + {x}^{3} + {x}^{2} + x + 1 = 0$ can be written as

$\dfrac{{x}^{5} - 1}{x - 1} = 0$$⇒ {x}^{5} = 1$

i.e. $x = \left(1\right)\dfrac{1}{5}$

So $\alpha, \beta, \gamma$ and $\delta$ are the ${5}^{\text{th}}$ roots of unity except $1$.

Now, from the concept of roots of unity, we know

{\alpha}^{n} + {\beta}^{n} + {\gamma}^{n} + {\delta}^{n} + {1}^{n} = \left\{\begin{cases}0 & \text{if}n\text{is not a multiple of}5 \\ 5 & \text{if}n\text{ is a multiple of}5\end{cases}\right)</p><p>So,{\alpha}^{2021} + {\beta}^{2021} + {\gamma}^{2021} + {\delta}^{2021} = - 1.