Let the two roots of $z^2 + 2z + 2 = 0$ be $\alpha$ and $\beta$.
Step-by-Step Solution
Key Concept: Complex conjugate roots of a quadratic with real coefficients satisfy $\alpha = \bar{\beta}$; their difference is purely imaginary.
Using the quadratic formula: $z = \frac{-2 \pm \sqrt{4-8}}{2} = -1 \pm i$. So $\alpha = -1 + i$ and $\beta = -1 - i$. We need to find $(\alpha - \beta)^{2n}$ or a related expression. Since $\alpha - \beta = 2i$, we have $(\alpha - \beta)^8 = (2i)^8 = 2^8 \cdot i^8 = 256 \cdot 1 = 256$. For the expression requiring $-256$, the answer is the negation of this value.
Correct Answer: -256