To factorize the quadratic expression \( x^2 + 6xy + 9y^2 \), we can recognize this as a perfect square trinomial. A perfect square trinomial is of the form \( a^2 + 2ab + b^2 \), which can be factored as \( (a + b)^2 \).

Here, let's identify the components:

- \( a^2 = x^2 \), so \( a = x \).
- \( b^2 = 9y^2 \), so \( b = 3y \).
- The middle term \( 6xy \) is twice the product of \( x \) and \( 3y \), which matches \( 2ab = 2(x)(3y) \).

Thus, we can factor the expression as:
\[
x^2 + 6xy + 9y^2 = (x + 3y)^2
\]

So, the factorized form of \( x^2 + 6xy + 9y^2 \) is \( (x + 3y)^2 \).

Would you like more details or have any questions?

Here are five related questions to expand your understanding:
1. How do you factorize expressions that are not perfect square trinomials?
2. Can you factorize \( x^2 + 4xy + 4y^2 \)?
3. What is the general process to identify a perfect square trinomial?
4. How can you expand \( (x + 3y)^2 \) to check the factorization?
5. What are some real-world applications of factoring quadratic expressions?

**Tip:** Recognizing patterns like perfect square trinomials can significantly speed up the factorization process.

Learn how to factorize the quadratic expression x^2 + 6xy + 9y^2 using the perfect square trinomial method. This problem covers key algebraic concepts, including perfect square trinomials and their factorization. Suitable for students in Grades 7-9.

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