What methods can be used to rewrite square trinomials and difference of squares binomials as separate factors?

1- As per the difference of squares:
Difference of squares is simply written in the form of (a² - b²) and to find its roots, you simply simplify the previous expression to be (a+b)(a-b) where a and b are the roots
i.e. (a² - b²) = (a+b)(a-b)

2- As per square trinomials:
Let's assume that the square trinomial is written in the form of (ax² + bx +c) where a,b and c are constants.
Now the basic formula to get the two roots of this expression is given as shown in the below picture.
This formula is valid for almost all expressions, nevertheless, there is a special case (perfect square) in which we do not have to use to use this formula.
Perfect Square: 
Let's assume that we have an expression written in the form of                 (ax² + cx +b) where a,b and c are constants. If you find that the value of a is the square of a first binomial, b is the square of a second binomial and c is the twice the value of multiplying a and b (i.e. c=2ab), then this expression is said to be a perfect square and can simply be simplified as (a+b)²
Note: If the expression of the perfect square was (ax² - cx +b), then the simplification would be (a-b)²

Wow. that is a really cool question.