What's the fastest approach?

Let $a=x$ , then ${ x }^{ 2 }-6x-2={ a }^{ 2 }-6a-2=0$ . Multiply the expression by ${ a }^{ 8 }$ . Thus,

${ a }^{ 10 }-6{ a }^{ 9 }-2{ a }^{ 8 }=0\\ \Rightarrow \frac { { a }^{ 10 }-2{ a }^{ 8 } }{ 2{ a }^{ 9 } } =3$

Let $a_n={\alpha}^n+{\beta}^n$ . From the equation we get that $\alpha+\beta=a_1=6,{\alpha}{\beta}=-2$ using vieta's formula.

$6 a_{n-1}=(\alpha+\beta) ({\alpha}^{n-1}+{\beta}^{n-1})={\alpha}^n+{\beta}^n+ {\alpha}{\beta}({\alpha}^{n-2}+{\beta}^{n-2}))=a_n-2a_{n-2}$

So $6 a_{n-1}=a_n-2 a_{n-2} \Rightarrow 6 a_9=a_{10}-2a_8$

$\frac{a_{10}-2 a_{8}}{2 a_9}=\frac{6 a_9}{2 a_9}=3$