Let $\alpha$ and $\beta$ be two roots of the equation $x^{2} + 2x + 2 = 0$, then $\alpha^{15} + \beta^{15}$ is equal to

Roots of x^2 + 2x + 2 = 0 are x = (-2 +/- sqrt(4-8))/2 = -1 +/- i. In polar form, -1+i = sqrt(2) * (cos(3pi/4) + i*sin(3pi/4)). By De Moivre's Theorem, alpha^15 + beta^15 = 2 * (sqrt(2))^15 * cos(15 * 3pi/4) = 2 * 2^7.5 * cos(45pi/4) = 2 * 2^7.5 * cos(11pi + pi/4) = 2 * 2^7.5 * (-cos(pi/4)) = 2 * 2^7.5 * (-1/sqrt(2)) = -2 * 2^7 = -256.