Let alpha and beta be two roots of the equation

Consider the equation

$x^{2}+2 x+2=0$

$x=\frac{-2 \pm \sqrt{4-8}}{2}=-1 \pm i$

Let $\alpha=-1+i, \beta=-1-i$

$\alpha^{15}+\beta^{15}=(-1+i)^{15}+(-1-i)^{15}$

$=\left(\sqrt{2} e^{l \frac{3 \pi}{4}}\right)^{15}+\left(\sqrt{2} e^{-i \frac{3 \pi}{4}}\right)^{15}$

$=(\sqrt{2})^{15}\left[e^{\frac{i 45 \pi}{4}}+e^{\frac{-i 45 \pi}{4}}\right]$

$=(\sqrt{2})^{15} \cdot 2 \cos \frac{45 \pi}{4}=(\sqrt{2})^{15} \cdot 2 \cos \frac{3 \pi}{4}$

$==\frac{-2}{\sqrt{2}}(\sqrt{2})^{15}$

$=-2(\sqrt{2})^{14}=-256$