G(x+3)=x^2+2x+1

Solution for G(x+3)=x^2+2x+1 equation:

(G+3)=G^2+2G+1

We move all terms to the left:

(G+3)-(G^2+2G+1)=0

We get rid of parentheses

-G^2+G-2G+3-1=0

We add all the numbers together, and all the variables

-1G^2-1G+2=0

a = -1; b = -1; c = +2;
Δ = b2-4ac
Δ = -12-4·(-1)·2
Δ = 9
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$G_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$G_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{9}=3$
$G_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1)-3}{2*-1}=\frac{-2}{-2}
=1
$
$G_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+3}{2*-1}=\frac{4}{-2}
=-2
$