F(x+1)=2x^2+8x-6

Solution for F(x+1)=2x^2+8x-6 equation:

(F+1)=2F^2+8F-6

We move all terms to the left:

(F+1)-(2F^2+8F-6)=0

We get rid of parentheses

-2F^2+F-8F+1+6=0

We add all the numbers together, and all the variables

-2F^2-7F+7=0

a = -2; b = -7; c = +7;
Δ = b2-4ac
Δ = -72-4·(-2)·7
Δ = 105
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-7)-\sqrt{105}}{2*-2}=\frac{7-\sqrt{105}}{-4}
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-7)+\sqrt{105}}{2*-2}=\frac{7+\sqrt{105}}{-4}
$