Hi Xibu4. Good. You also know that
α + β = 0
and
γ + δ = -5/6
Solve those equations for β and δ, and then substitute the results in
αβ + γδ = −4
So, you've already used the formula for sum of root products taken two at a time, for a polynomial Ax
4+Bx
3+Cx
2+Dx+E:
αβ + βγ + γδ + δα + αγ + βδ = C/A
Have you seen the formula for sum of root products taken three at a time?
αβγ + βγδ + γδα + αβδ = -D/A
Again, make the same substitutions for β and δ.
You now ought to have one equation containing α only, and another equation containing α and γ. Solve that system.