To factor the quadratic expression \( 15x^2 + 22x + 8 \), we can use factoring by grouping. Here's the step-by-step process:

### Step 1: Multiply the coefficient of \(x^2\) (15) and the constant term (8):
\[
15 \times 8 = 120
\]

### Step 2: Find two numbers that multiply to 120 and add up to the middle coefficient, 22. These numbers are 10 and 12 because:
\[
10 \times 12 = 120 \quad \text{and} \quad 10 + 12 = 22
\]

### Step 3: Rewrite the middle term (22x) using 10x and 12x:
\[
15x^2 + 10x + 12x + 8
\]

### Step 4: Factor by grouping:
\[
5x(3x + 2) + 4(3x + 2)
\]

### Step 5: Factor out the common binomial \((3x + 2)\):
\[
(5x + 4)(3x + 2)
\]

So, the factored form of \( 15x^2 + 22x + 8 \) is:
\[
(5x + 4)(3x + 2)
\]

Would you like further details or have any questions?

Here are 5 related questions for you to consider:
1. How can you apply factoring by grouping to more complex expressions?
2. What is the importance of recognizing the middle term in factorization?
3. How would this factorization change if the signs of the terms were different?
4. How can the distributive property verify the factored form?
5. What are some alternative methods to factor quadratic expressions?

**Tip:** When factoring, always check if there's a greatest common factor (GCF) before proceeding with grouping or other methods.

Learn how to factor the quadratic expression 15x² + 22x + 8 using factoring by grouping. This solution explains each step in detail and applies the distributive property to verify the result. Suitable for students in Grades 8-10.

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