F(x+1)=x^2+9x+4

Solution for F(x+1)=x^2+9x+4 equation:

(F+1)=F^2+9F+4

We move all terms to the left:

(F+1)-(F^2+9F+4)=0

We get rid of parentheses

-F^2+F-9F+1-4=0

We add all the numbers together, and all the variables

-1F^2-8F-3=0

a = -1; b = -8; c = -3;
Δ = b2-4ac
Δ = -82-4·(-1)·(-3)
Δ = 52
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:

$\sqrt{\Delta}=\sqrt{52}=\sqrt{4*13}=\sqrt{4}*\sqrt{13}=2\sqrt{13}$
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-2\sqrt{13}}{2*-1}=\frac{8-2\sqrt{13}}{-2}
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+2\sqrt{13}}{2*-1}=\frac{8+2\sqrt{13}}{-2}
$