We are given the expression:

$36x^2 - 9(x - y)^2$

Step-by-step factorization:

1. Recognize a common factor: Both terms in the expression contain a factor of $9$. We can factor $9$ out of the entire expression:

   $36x^2 - 9(x - y)^2 = 9[4x^2 - (x - y)^2]$

2. Apply the difference of squares: The expression inside the brackets, $4x^2 - (x - y)^2$, is a difference of squares. The difference of squares formula is:

   $a^2 - b^2 = (a - b)(a + b)$

   Here, $a = 2x$ and $b = (x - y)$, so applying the difference of squares:

   $4x^2 - (x - y)^2 = (2x - (x - y))(2x + (x - y))$

   Simplifying both factors: $2x - (x - y) = 2x - x + y = x + y$ $2x + (x - y) = 2x + x - y = 3x - y$

3. Final factorized form: Now, substituting these simplified expressions back into the factorization:

   [ 9[4x^2 - (x - y)^2] = 9[(x + y)(3x - y)] ]

Thus, the fully factorized form is:

$9(x + y)(3x - y)$

Would you like any more details on these steps or further explanations?

Related Questions:

1. What is the difference of squares formula, and how is it applied?
2. How do you factor trinomials using grouping?
3. Can you factor more complex expressions using the difference of cubes?
4. What are the steps for factoring a quadratic expression?
5. How do you recognize special factoring patterns in polynomial expressions?

Tip:

Always check for a greatest common factor (GCF) before proceeding with more advanced factoring techniques like the difference of squares or trinomials!