To determine which expression is equivalent to the given quadratic expression [tex]\( 2x^2 - 11x - 6 \)[/tex], we will factorize the quadratic expression step-by-step.
### Step-by-Step Factorization
1. Identify the quadratic expression:
[tex]\[
2x^2 - 11x - 6
\][/tex]
2. Find two numbers that multiply to the product of the coefficient of [tex]\(x^2\)[/tex] term (which is 2) and the constant term (which is -6):
[tex]\[
2 \times -6 = -12
\][/tex]
We need two numbers that multiply to -12 and add to the coefficient of the [tex]\(x\)[/tex] term, which is -11.
3. Determine the two numbers:
Consider the two numbers: -12 and 1
[tex]\[
-12 \times 1 = -12 \quad \text{and} \quad -12 + 1 = -11
\][/tex]
Thus, the two numbers are -12 and 1.
4. Rewrite the middle term (-11x) using -12 and 1:
[tex]\[
2x^2 - 12x + x - 6
\][/tex]
5. Factor by grouping:
Group the terms in pairs and factor out the common factors:
[tex]\[
2x(x - 6) + 1(x - 6)
\][/tex]
6. Factor out the common binomial factor:
[tex]\[
(2x + 1)(x - 6)
\][/tex]
So, the correct factorization of [tex]\(2x^2 - 11x - 6\)[/tex] is [tex]\((2x + 1)(x - 6)\)[/tex].
### Verify by expanding
Let's verify our factorization by expanding [tex]\((2x + 1)(x - 6)\)[/tex]:
[tex]\[
(2x + 1)(x - 6) = 2x(x - 6) + 1(x - 6) = 2x^2 - 12x + x - 6 = 2x^2 - 11x - 6
\][/tex]
The obtained expression matches the given expression, confirming our factorization is correct.
Therefore, the correct answer is:
A. [tex]\((2x + 1)(x - 6)\)[/tex]
