$$ \begin{aligned} & \alpha^2-6 \alpha-2=0 \\ & \alpha^{10}-6 \alpha 9-2 \alpha^8=0 \end{aligned} $$ Similarly $\beta^{10}-6 \beta^9-2 \beta^8=0$ $$ \begin{aligned} & \left(\alpha^{10}-\beta^{10}\right)-6\left(\alpha^9-\beta^9\right)-2\left(\alpha^8-\beta^8\right)=0 \\ & \Rightarrow \quad a_{10}-6 a_9-2 a_8=0 \\ & \quad \quad \frac{a_{10}-2 a_8}{3 a_9}=2 \\ & \Rightarrow \quad \end{aligned} $$