The expression \(9x^2 + 66xy + 121y^2\) is a quadratic trinomial. To factor it, we'll use the following steps:

### Step 1: Check if it’s a perfect square trinomial.
A trinomial of the form \(a^2 + 2ab + b^2\) factors as \((a + b)^2\).

Here, observe that:

- \(9x^2\) is a perfect square because \(\sqrt{9x^2} = 3x\).
- \(121y^2\) is a perfect square because \(\sqrt{121y^2} = 11y\).
- Check if the middle term \(66xy\) equals \(2 \times 3x \times 11y\):
  
  \[
  2 \times 3x \times 11y = 66xy
  \]

Thus, it is indeed a perfect square trinomial.

### Step 2: Write the factorization.
Now that we've verified it's a perfect square trinomial, we can factor it as:

\[
9x^2 + 66xy + 121y^2 = (3x + 11y)^2
\]

So, the factorization of \(9x^2 + 66xy + 121y^2\) is:

\[
(3x + 11y)^2
\]

Would you like more details or have any questions? 

Here are 5 related questions to consider:
1. How do you factor non-perfect square trinomials?
2. Can perfect square trinomials have negative middle terms?
3. What are the steps to factor a difference of squares?
4. How do you factor a quadratic trinomial with a non-perfect square leading coefficient?
5. Can a trinomial be factored if it’s not a perfect square?

**Tip**: Recognizing patterns like perfect square trinomials helps you factor efficiently!

Learn how to factor the quadratic trinomial 9x^2 + 66xy + 121y^2 using the perfect square trinomial method. This problem covers key algebraic concepts, including identifying perfect squares and applying the formula for a^2 + 2ab + b^2. Suitable for students in Grades 8-10.

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