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

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

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

(By multiplying $\left.\alpha^{n}\right)$

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

By adding (1)$\&(2)$

$5 \mathrm{S}_{\mathrm{n}+2}+6 \mathrm{S}_{\mathrm{n}+1}-2 \mathrm{S}_{\mathrm{n}}=0$

For $n=4$

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