To factor the quadratic expression [tex]\(8x^2 + 13x - 6\)[/tex], we need to break it down into simpler binomials.
Let's detail the steps to factorize the expression [tex]\(8x^2 + 13x - 6\)[/tex]:
### Step 1: Identify the coefficients
The given quadratic expression is [tex]\(8x^2 + 13x - 6\)[/tex], where:
- [tex]\(a = 8\)[/tex] (coefficient of [tex]\(x^2\)[/tex]),
- [tex]\(b = 13\)[/tex] (coefficient of [tex]\(x\)[/tex]),
- [tex]\(c = -6\)[/tex] (constant term).
### Step 2: Set up the factor pairs
We need to find two numbers that multiply to [tex]\(a \cdot c = 8 \cdot (-6) = -48\)[/tex] and add up to [tex]\(b = 13\)[/tex].
### Step 3: Find the pair of numbers
The pair of numbers that fit these criteria are:
- 16 (positive) and -3 (negative).
These numbers multiply to [tex]\(16 \cdot (-3) = -48\)[/tex] and add to [tex]\(16 + (-3) = 13\)[/tex].
### Step 4: Split the middle term
We rewrite the middle term [tex]\(13x\)[/tex] using the pair of numbers found:
[tex]\[8x^2 + 16x - 3x - 6.\][/tex]
### Step 5: Factor by grouping
Group the terms to factor them separately:
[tex]\[(8x^2 + 16x) + (-3x - 6).\][/tex]
Factor out the common factors in each group:
[tex]\[8x(x + 2) - 3(x + 2).\][/tex]
### Step 6: Factor out the common binomial
Notice that [tex]\((x + 2)\)[/tex] is a common binomial term:
[tex]\[(8x - 3)(x + 2).\][/tex]
Thus, the factorization of [tex]\(8x^2 + 13x - 6\)[/tex] is:
[tex]\[(x + 2)(8x - 3).\][/tex]
Therefore, among the given choices, the correct factorization is:
[tex]\[(x + 2)(8x - 3).\][/tex]
