Question:

Factor 4x² + 12x + 9 - 49w²

Do you see this is the difference of two squares?  One of the squares is a perfect square trinomial.

(4x² + 12x + 9) - (49w²)

Now, let's express each part of the expression as a perfect square.  First, the perfect square trinomial:
(4x² + 12x + 9) is factored as (2x + 3)².  Now the other part of the expression:
(49w²) can be expressed as (7w)².  Now this expression becomes...

(2x + 3)² - (7w)² 

Clearly this is the difference of two squares.  Now let's apply what we learned before -- that a² - b² can be factored as (a + b) (a - b).  In this case, our expression is factored as:

((2x + 3) + (7w)) ((2x + 3) - (7w))

Now, removing extra parentheses, the fully factored expression is this:

(2x + 3 + 7w) (2x + 3 - 7w)