The given expression is:

\[
x^2 - 6x + 9 - 4y^2
\]

This can be simplified or rewritten by recognizing patterns.

First, notice that the first three terms form a perfect square trinomial:
\[
x^2 - 6x + 9 = (x - 3)^2
\]

So, the expression can be rewritten as:
\[
(x - 3)^2 - 4y^2
\]

This is now in the form of a **difference of squares**, which follows the formula:
\[
a^2 - b^2 = (a - b)(a + b)
\]

Here, \( a = (x - 3) \) and \( b = 2y \), so applying the formula, we get:
\[
(x - 3)^2 - (2y)^2 = (x - 3 - 2y)(x - 3 + 2y)
\]

Thus, the factored form of the expression is:
\[
(x - 3 - 2y)(x - 3 + 2y)
\]

### Would you like more details or have any questions about this process?

Here are 5 related questions to expand your understanding:
1. How do you identify a perfect square trinomial?
2. What other algebraic identities can be used to factor expressions?
3. Can you expand and simplify the factored expression to verify it's equivalent to the original?
4. How does the difference of squares help in simplifying complex algebraic expressions?
5. What happens if the signs of the \( y^2 \)-term change?

**Tip:** Always check for patterns like perfect squares and differences of squares, as they make factoring easier.

Discover how to factor the algebraic expression x^2 - 6x + 9 - 4y^2 using perfect square trinomials and the difference of squares theorem. Learn step-by-step how to simplify and factor this expression, suitable for Grades 8-10.

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