Let alpha and beta be the roots of the equation

$\alpha$ and $\beta$ are roots of $5 x^{2}+6 x-2=0$

$\Rightarrow 5 \alpha^{2}+6 \alpha-2=0$

$\Rightarrow 5 \alpha^{n+2}+6 \alpha^{n+1}-2 \alpha^{n}=0$.......(1)

(By multiplying $\alpha^{\mathrm{n}}$ )

Similarly $5 \beta^{n+2}+6 \beta^{n+1}-2 \beta^{n}=0$$\ldots(2)$

By adding (1) & (2)

$5 S_{n+2}+6 S_{n+1}-2 S_{n}=0$

For $n=4$

$5 \mathrm{~S}_{6}+6 \mathrm{~S}_{5}=2 \mathrm{~S}_{4}$