$\begin{aligned} & x^2+2 x+2=0 \\ & x= \frac{-2 \pm \sqrt{4-8}}{2}=\frac{-2 \pm 2 i}{2} \\ & x=-1 \pm i, \alpha=-1+i, \beta=-1-i \\ & \alpha^{15}+\beta^{15}=(-1+i)^{15}+(-1-i)^{15} \\ &=(\sqrt{2})^{15}\left[\left(\frac{-1}{\sqrt{2}}+\frac{i}{\sqrt{2}}\right)^{15}-\left(\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}}\right)^{15}\right] \\ &=(\sqrt{2})^{15}\left[\left(\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}\right)^{15}-\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)^{15}\right] \\ &=(\sqrt{2})^{15}\left[\left(e^{i 3 \pi / 4}\right)^{15}-\left(e^{i \pi / 4}\right)^{15}\right] \\ &=(\sqrt{2})^{15}\left[e^{i 45 \pi / 4}-e^{i 15 \pi / 4}\right] \\ &=(\sqrt{2})^{15}\left[e^{i\left(\frac{5 \pi}{4}\right)}-e^{i\left(\frac{-\pi}{4}\right)}\right] \\ &=(\sqrt{2})^{15}\left[\cos \frac{5 \pi}{4}+i \sin \frac{5 \pi}{4}-\cos \left(\frac{-\pi}{4}\right)-i \sin \left(\frac{-\pi}{4}\right)\right] \\ &=(\sqrt{2})^{15}\left[-\cos \frac{\pi}{4}-\cos \frac{\pi}{4}\right]=(\sqrt{2})^{15}(-\sqrt{2})=-2^8 .\end{aligned}$