Solve similar right triangles find lenght AE issue

(This looks to me like a question that you could ask within the lesson you found it in (since you are just using text), and maybe get a faster, better answer.)

It looks like this quadratic equation is being solved by completing the square.
I recommend the Algebra 1 article from Unit 13, Lesson 8 "Factoring quadratics: Perfect squares".

Completing the square depends on looking at the coefficients of the quadratic term and the linear term, then calculating what constant term needs to be added to complete the square.
A simple example would be:
y² + 6y = 7
We know (y + 3)² = y² + 6y + 9, so we add 9 to both sides:
y² + 6y + 9 = 7 + 9
(y + 3)² = 16

In general, y² + by + (b/2)² = (y + b/2)², so add (b/2)² to both sides.
Does this make sense?

In (AE)² + 8.4(AE) = 576, b=8.4, so we add (8.4/2)² to both sides:
(AE)² + 2(4.2)(AE) + (4.2)² = 576 + (4.2)²

The left side is already in the form of an expanded square, so it can be condensed into the square; on the right side, 4.2² = 17.64, so 576 + 4.2² = 576 + 17.64 = 593.64:
(AE + 4.2)² = 593.64