Let c and b be the roots of x²-6x-2=0 (cb). If

a_n=cⁿ-bⁿ where (n=1)

then a₁₀-2a₈/2a₉ =

1. 1
2. 2
3. 3
4. 4

c and b are the roots of the equation : $x^2-6x-2=0 (c>b)$............(1)
if $a_n=c^n-b^n (n\ge 1)$
c is the root of eq(1): therefore 
$$\begin{gathered} c^2-6c-2=0 \\ \Rightarrow c^2=6c+2.........\left(2\right) \end{gathered}$$
multiplying through out by  $c^{n-2}$ on the both sides:
$c^n=6c^{n-1}+2c^{n-2}.........\left(3\right)$
similarly $b^n=6b^{n-1}+2b^{n-2}..........\left(4\right)$
subtracting eq(4) by eq(3):
$$\begin{gathered} c^n-b^n=6(c^{n-1}-b^{n-1})+2(c^{n-2}-b^{n-2}) \\ {a}_n=6a_{n-1}+2a_{n-2}........\left(5\right) \end{gathered}$$
now put n = 10
$a_{10}=6a_9+2a_8$
therefore $a_{10}-2a_8=6a_9$
hence
$\frac{a_{10}-2a_8}{2a_9}=\frac{6a_9}{2a_9}=3$
hope this helps you