To determine which expression is equivalent to the given quadratic expression [tex]\(2x^2 - 11x - 6\)[/tex], we will compare each option with the factored form of the expression.
### Step-by-Step Solution:
1. Factor the given quadratic expression:
- Given quadratic expression: [tex]\(2x^2 - 11x - 6\)[/tex].
- To factor [tex]\(2x^2 - 11x - 6\)[/tex], we look for two numbers that multiply to [tex]\(2 \times -6 = -12\)[/tex] and add to [tex]\(-11\)[/tex].
- The two numbers are 1 and -12, since [tex]\(1 \cdot -12 = -12\)[/tex] and [tex]\(1 + (-12) = -11\)[/tex].
- Rewrite the middle term [tex]\(-11x\)[/tex] using these two numbers: [tex]\(2x^2 + x - 12x - 6\)[/tex].
- Group the terms: [tex]\((2x^2 + x) + (-12x - 6)\)[/tex].
- Factor by grouping: [tex]\(x(2x + 1) - 6(2x + 1)\)[/tex].
- Factor out the common binomial factor: [tex]\((2x + 1)(x - 6)\)[/tex].
Thus, the factored form of [tex]\(2x^2 - 11x - 6\)[/tex] is [tex]\((2x + 1)(x - 6)\)[/tex].
2. Compare with each option:
- Option A: [tex]\(2(x - 3)(x + 1)\)[/tex]
- [tex]\(= 2(x^2 - 2x - 3)\)[/tex]
- [tex]\(= 2x^2 - 4x - 6\)[/tex].
- This does not match [tex]\(2x^2 - 11x - 6\)[/tex].
- Option B: [tex]\(2(x + 3)(x - 2)\)[/tex]
- [tex]\(= 2(x^2 + x - 6)\)[/tex]
- [tex]\(= 2x^2 + 2x - 12\)[/tex].
- This does not match [tex]\(2x^2 - 11x - 6\)[/tex].
- Option C: [tex]\((2x + 1)(x - 6)\)[/tex]
- Multiply to check:
- [tex]\( (2x + 1)(x - 6) = 2x(x - 6) + 1(x - 6) \)[/tex]
- [tex]\(= 2x^2 - 12x + x - 6\)[/tex]
- [tex]\(= 2x^2 - 11x - 6\)[/tex].
- This matches exactly.
- Option D: [tex]\((2x + 3)(x - 2)\)[/tex]
- Multiply to check:
- [tex]\( (2x + 3)(x - 2) = 2x(x - 2) + 3(x - 2) \)[/tex]
- [tex]\(= 2x^2 - 4x + 3x - 6\)[/tex]
- [tex]\(= 2x^2 - x - 6\)[/tex].
- This does not match [tex]\(2x^2 - 11x - 6\)[/tex].
Based on the comparisons, the correct expression that is equivalent to the given expression [tex]\(2x^2 - 11x - 6\)[/tex] is:
[tex]\[
\boxed{(2x + 1)(x - 6)}
\][/tex]
So, the correct answer is Option C: [tex]\((2x + 1)(x - 6)\)[/tex].
