F(a+1)=4a^2+2a-10

Solution for F(a+1)=4a^2+2a-10 equation:

(F+1)=4F^2+2F-10

We move all terms to the left:

(F+1)-(4F^2+2F-10)=0

We get rid of parentheses

-4F^2+F-2F+1+10=0

We add all the numbers together, and all the variables

-4F^2-1F+11=0

a = -4; b = -1; c = +11;
Δ = b2-4ac
Δ = -12-4·(-4)·11
Δ = 177
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{-(-1)-\sqrt{177}}{2*-4}=\frac{1-\sqrt{177}}{-8}
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+\sqrt{177}}{2*-4}=\frac{1+\sqrt{177}}{-8}
$