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

$$\begin{gathered} {(x+1)}^2+1=0 {(x+1)}^2-{\left(\text{i}\right)}^2=0 (x+1+\text{i})(\text{x+1}-\text{i})=0 \\ \therefore x=\frac{-(1+i)}{\alpha \left(let\right)},\frac{-(1-i)}{\beta \left(let\right)} So, {\alpha }^{15}+{\beta }^{15}={\left({\alpha }^2\right)}^7\alpha +{\left({\beta }^2\right)}^7\beta \\ =-128(-i+1+i+1) =-256 \end{gathered}$$