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The expression to be factored is:

\[
3x^2 + 10x - 8
\]

We need to factor it into the form:

\[
( \text{?}x - [\text{?}] )( x + [\text{?}] )
\]

Here’s the step-by-step process:

1. **Identify the coefficients**: 
   - \( a = 3 \) (coefficient of \(x^2\)),
   - \( b = 10 \) (coefficient of \(x\)),
   - \( c = -8 \) (constant term).

2. **Multiply \( a \) and \( c \)**: 
   - \( 3 \times -8 = -24 \).

3. **Find two numbers that multiply to -24 and add to 10**:
   - The pair of numbers is \( 12 \) and \( -2 \) because:
     \[
     12 \times -2 = -24
     \quad \text{and} \quad 12 + (-2) = 10
     \]

4. **Rewrite the middle term (10x) using 12x and -2x**:
   \[
   3x^2 + 12x - 2x - 8
   \]

5. **Factor by grouping**:
   - Group the terms: 
     \[
     (3x^2 + 12x) - (2x + 8)
     \]
   - Factor each group:
     \[
     3x(x + 4) - 2(x + 4)
     \]
   - Now, factor out the common binomial factor \((x + 4)\):
     \[
     (3x - 2)(x + 4)
     \]

Thus, the factorization of \( 3x^2 + 10x - 8 \) is:

\[
(3x - 2)(x + 4)
\]

Would you like more details or have any questions?

---

**Questions to expand understanding:**
1. How can you verify that the factorization is correct?
2. What is the importance of multiplying \(a \times c\) in factoring trinomials?
3. What happens when you cannot find integers that fit the middle term?
4. Can this method be used for all quadratic equations? Why or why not?
5. How does factoring help in solving quadratic equations?

**Tip:** When factoring trinomials, always check if you can factor out a common factor first before proceeding with other methods.

Learn how to factor the quadratic equation 3x^2 + 10x - 8 using a step-by-step process. This example covers key algebraic concepts, including factoring by grouping and the zero product property. Suitable for students in Grades 8-10.

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