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Let's factor the quadratic expression \( 3x^2 + 24x + 45 \) completely.

### Step 1: Factor out the greatest common factor (GCF).
The GCF of the terms \( 3x^2 \), \( 24x \), and \( 45 \) is 3. So, we can factor out 3 first:

\[
3(x^2 + 8x + 15)
\]

### Step 2: Factor the quadratic inside the parentheses.
Now, we need to factor the trinomial \( x^2 + 8x + 15 \). We are looking for two numbers that multiply to 15 and add to 8.

The numbers 3 and 5 work because:
\[
3 \times 5 = 15 \quad \text{and} \quad 3 + 5 = 8
\]

### Step 3: Write the factors.
The factorization of \( x^2 + 8x + 15 \) is \( (x + 3)(x + 5) \).

### Final Answer:
Therefore, the complete factorization of \( 3x^2 + 24x + 45 \) is:

\[
3(x + 3)(x + 5)
\]

Would you like more details or have any questions?

Here are some related questions:
1. How can we verify the factorization of a quadratic expression?
2. What other methods can be used to factor quadratics?
3. How do you handle factoring when the leading coefficient is greater than 1?
4. What happens when a quadratic is not factorable?
5. How can we apply factoring to solve quadratic equations?

**Tip:** Always check your factorization by expanding the factors back to confirm the original expression!

Learn how to factor the quadratic expression 3x^2 + 24x + 45 step-by-step, using the greatest common factor and trinomial factoring. This solution is ideal for students in Grades 8-10, covering algebraic techniques for factoring quadratics.

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