Since $\alpha$ and $\beta$ are the roots(solutions) of the equation

$x^2-6 x-2=0$

So, it will satisfy the equation

$\alpha^2-6 \alpha-2=0$

$\beta^2-6 \beta-2=0$

Now, we have to find the value of $\frac{a_{10}-2 a_8}{2 a_9}$

Given, $a_n=\alpha^n-\beta^n$

So, $a_{10}=\alpha^{10}-\beta^{10}$

So, to get the values of $\alpha^{10}$ and $\beta^{10}$, multiplying equation $(1)$ and $(2)$ by $\alpha^8$ and $\beta^8$ respectively.

$\alpha^{10}-6 \alpha^9-2 \alpha^8=0$

$\text { or, } \alpha^{10}=6 \alpha^9+2 \alpha^8 \ldots . . \text { (4) }$

$\beta^{10}-6 \beta^9-2 \beta^8=0$

$\text { or, } \beta^{10}=6 \beta^9+2 \beta^8 \ldots . . \text { (5) }$

So, using equation$(4)$ and $(5)$ in $(3)$, we get

$a_{10}=6 \alpha^9+2 \alpha^8-\left(6 \beta^9+2 \beta^8\right)$

$a_{10}=6 \alpha^9+2 \alpha^8-6 \beta^9-2 \beta^8$

$=6\left(\alpha^9-\beta^9\right)+2\left(\alpha^8-\beta^8\right)$

$a_{10}=6 a_9+2 a_8$

$a_{10}-2 a_8=6 a_9$

$\frac{a_{10}-2 a_8}{2 a_9}=3$