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Let's solve the equation \( 9x^2 + 24x + 16 \) by factoring it.

We will factor this quadratic expression using the following steps:

### 1. Express the quadratic equation:
\[ 9x^2 + 24x + 16 \]

### 2. Identify if it's a perfect square trinomial:
A trinomial of the form \( a^2 + 2ab + b^2 \) can be factored as \( (a + b)^2 \).

Here, we notice:
- \( 9x^2 \) is a perfect square, which is \( (3x)^2 \).
- \( 16 \) is also a perfect square, which is \( 4^2 \).
- \( 24x \) fits the pattern \( 2ab \), where \( a = 3x \) and \( b = 4 \), because \( 2 \cdot 3x \cdot 4 = 24x \).

Thus, the expression is a perfect square trinomial, and it can be factored as:
\[
(3x + 4)^2
\]

### 3. Final Answer:
The correct factorization is \( (3x + 4)^2 \).

This matches option **d**.

Let me know if you'd like further details or have any questions!

Here are 5 related questions:
1. How do you factor general quadratic expressions that are not perfect squares?
2. What is the method to complete the square for any quadratic equation?
3. Can you always factor quadratics into real numbers?
4. How does factoring relate to solving quadratic equations?
5. What are some examples of trinomials that cannot be factored easily?

**Tip:** When checking for perfect squares in trinomials, always verify if both the first and last terms are perfect squares and check the middle term using \( 2ab \).

Which of the following is the correct factorization of 9x^2 + 24x + 16?

Learn how to factorize the quadratic equation 9x^2 + 24x + 16 into (3x + 4)^2, which is a perfect square trinomial. This algebraic problem covers essential concepts like factoring and identifying perfect square trinomials, suitable for Grades 7-9.

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