To factor the quadratic polynomial [tex]\( 3x^2 + 23x - 8 \)[/tex], we aim to find two binomials [tex]\((ax + b)(cx + d)\)[/tex] that multiply to give the original polynomial. Let’s break down the steps to solve this.
### Step-by-Step Solution:
1. Identify coefficients:
The polynomial is [tex]\( 3x^2 + 23x - 8 \)[/tex].
- [tex]\(a = 3\)[/tex] (coefficient of [tex]\(x^2\)[/tex])
- [tex]\(b = 23\)[/tex] (coefficient of [tex]\(x\)[/tex])
- [tex]\(c = -8\)[/tex] (constant term)
2. Set up the factor form:
We look for two numbers that multiply to [tex]\( a \cdot c = 3 \cdot (-8) = -24 \)[/tex] and add up to [tex]\( b = 23 \)[/tex].
3. Find the pair of numbers:
We need two numbers whose product is [tex]\(-24\)[/tex] and whose sum is [tex]\(23\)[/tex]. After considering possible pairs, we see that [tex]\(24\)[/tex] and [tex]\(-1\)[/tex] work:
- Product of [tex]\(24\)[/tex] and [tex]\(-1\)[/tex] is [tex]\(24 \cdot (-1) = -24\)[/tex].
- Sum of [tex]\(24\)[/tex] and [tex]\(-1\)[/tex] is [tex]\(24 + (-1) = 23\)[/tex].
4. Rewrite the middle term using these numbers:
Rewrite [tex]\(23x\)[/tex] as [tex]\(24x - x\)[/tex]:
[tex]\[
3x^2 + 23x - 8 = 3x^2 + 24x - x - 8
\][/tex]
5. Factor by grouping:
Group the terms to factor by grouping:
[tex]\[
= 3x(x + 8) - 1(x + 8)
\][/tex]
Notice that [tex]\((x + 8)\)[/tex] is a common factor:
[tex]\[
= (3x - 1)(x + 8)
\][/tex]
So the factors of [tex]\( 3x^2 + 23x - 8 \)[/tex] are [tex]\( (3x - 1)(x + 8) \)[/tex].
Therefore, the correct option is:
[tex]\[
(3x - 1)(x + 8)
\][/tex]
