Given, $\alpha , \beta$ are roots of $x^2-6x-2=0$.

$\Rightarrow {\alpha }^2-6\alpha -2=0$ and  ${\beta }^2-6\beta -2=0$

$\Rightarrow {\alpha }^2-2=6\alpha$ and${\beta }^2-2=6\beta ...\left(1\right)$

$\frac{a_{10}-2a_8}{2a_9}=\frac{\left({\alpha }^{10}-{\beta }^{10}\right)-2\left({\alpha }^8-{\beta }^8\right)}{2\left({\alpha }^9-{\beta }^9\right)}$

$=\frac{\left({\alpha }^{10}-2{\alpha }^8\right)-\left({\beta }^{10}-2{\beta }^8\right)}{2\left({\alpha }^9-{\beta }^9\right)}$

$=\frac{{\alpha }^8\left({\alpha }^2-2\right)-{\beta }^8\left({\beta }^2-2\right)}{2\left({\alpha }^9-{\beta }^9\right)}$

$=\frac{{\alpha }^8\left(6\alpha \right)-{\beta }^8\left(6\beta \right)}{2\left({\alpha }^9-{\beta }^9\right)}$       [$\because$ From (1)]

 $=\frac{6{\alpha }^9-6{\beta }^9}{2\left({\alpha }^9-{\beta }^9\right)}=3$