F(x+1)=5x^2+3x-6

Solution for F(x+1)=5x^2+3x-6 equation:

(F+1)=5F^2+3F-6

We move all terms to the left:

(F+1)-(5F^2+3F-6)=0

We get rid of parentheses

-5F^2+F-3F+1+6=0

We add all the numbers together, and all the variables

-5F^2-2F+7=0

a = -5; b = -2; c = +7;
Δ = b2-4ac
Δ = -22-4·(-5)·7
Δ = 144
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{144}=12$
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-12}{2*-5}=\frac{-10}{-10}
=1
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+12}{2*-5}=\frac{14}{-10}
=-1+2/5
$