$$\alpha $$ and $$\beta $$ be the roots of the equation
5x² + 6x – 2 = 0.

$$ \Rightarrow $$ 5$$\alpha $$² + 6$$\alpha $$ - 2 = 0

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

(By multiplying $$\alpha $$ⁿ)

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

By adding (1) & (2)

5S_n+2 + 6S_n+1 – 2S_n = 0

For n = 4

5S₆ + 6S₅ = 2S₄