Let's solve the problem step-by-step to find which expression is equivalent to [tex]\( 2x^2 - 11x - 6 \)[/tex].
We start by factoring the quadratic expression [tex]\( 2x^2 - 11x - 6 \)[/tex].
### Step 1: Write the quadratic expression
[tex]\[ 2x^2 - 11x - 6 \][/tex]
### Step 2: Set up the factors in the form [tex]\((ax + b)(cx + d)\)[/tex]
We know we need to find numbers [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex] such that the product of the binomials [tex]\((ax + b)(cx + d)\)[/tex] gives us the original quadratic expression when expanded.
### Step 3: Identify the correct factors
From our solution, we have:
[tex]\[ (x - 6)(2x + 1) \][/tex]
### Step 4: Expand the factors to verify
Let's expand [tex]\((x - 6)(2x + 1)\)[/tex]:
[tex]\[ (x - 6)(2x + 1) = x(2x + 1) - 6(2x + 1) \][/tex]
[tex]\[ = 2x^2 + x - 12x - 6 \][/tex]
[tex]\[ = 2x^2 - 11x - 6 \][/tex]
This is indeed the original quadratic expression. Therefore, the correct answer is:
[tex]\[ (x - 6)(2x + 1) \][/tex]
### Step 5: Match with the given options
Matching this with the options given:
- A. [tex]\(2(x - 3)(x + 1) \)[/tex] - Incorrect
- B. [tex]\(2(x + 3)(x - 2) \)[/tex] - Incorrect
- C. [tex]\((2x + 1)(x - 6) \)[/tex] - Correct
- D. [tex]\((2x + 3)(x - 2) \)[/tex] - Incorrect
So, the correct answer is:
C. [tex]\((2x + 1)(x - 6)\)[/tex]
