F(x+2)=4x^2+3x-3

Solution for F(x+2)=4x^2+3x-3 equation:

(F+2)=4F^2+3F-3

We move all terms to the left:

(F+2)-(4F^2+3F-3)=0

We get rid of parentheses

-4F^2+F-3F+2+3=0

We add all the numbers together, and all the variables

-4F^2-2F+5=0

a = -4; b = -2; c = +5;
Δ = b2-4ac
Δ = -22-4·(-4)·5
Δ = 84
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{84}=\sqrt{4*21}=\sqrt{4}*\sqrt{21}=2\sqrt{21}$
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-2\sqrt{21}}{2*-4}=\frac{2-2\sqrt{21}}{-8}
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+2\sqrt{21}}{2*-4}=\frac{2+2\sqrt{21}}{-8}
$