Which of the following is equivalent to `2x^2-6x-8`?

Options

A. `2(x-4)(x+1)`
B. 3(x+4)(x-1)
C. 2(x-3)(x+2)
D. 3(x-4)(x-2)

Answer

To determine which of the following expressions is equivalent to \(2x^2 - 6x - 8\), we will evaluate each option step by step.

### Step 1: Evaluate the First Option
The first option is \(2(x - 4)(x + 1)\).

1. **Expand the expression**:
\[ (x - 4)(x + 1) = x^2 + x - 4x - 4 = x^2 - 3x - 4 \]

2. **Multiply by 2**:
\[ 2(x^2 - 3x - 4) = 2x^2 - 6x - 8 \]

3. **Conclusion**: This matches the original expression \(2x^2 - 6x - 8\).

### Step 2: Evaluate the Second Option
The second option is \(3(x + 4)(x - 1)\).

1. **Expand the expression**:
\[ (x + 4)(x - 1) = x^2 - x + 4x - 4 = x^2 + 3x - 4 \]

2. **Multiply by 3**:
\[ 3(x^2 + 3x - 4) = 3x^2 + 9x - 12 \]

3. **Conclusion**: This does not match the original expression.

### Step 3: Evaluate the Third Option
The third option is \(2(x - 3)(x + 2)\).

1. **Expand the expression**:
\[ (x - 3)(x + 2) = x^2 + 2x - 3x - 6 = x^2 - x - 6 \]

2. **Multiply by 2**:
\[ 2(x^2 - x - 6) = 2x^2 - 2x - 12 \]

3. **Conclusion**: This does not match the original expression.

### Step 4: Evaluate the Fourth Option
The fourth option is \(3(x - 4)(x - 2)\).

1. **Expand the expression**:
\[ (x - 4)(x - 2) = x^2 - 2x - 4x + 8 = x^2 - 6x + 8 \]

2. **Multiply by 3**:
\[ 3(x^2 - 6x + 8) = 3x^2 - 18x + 24 \]

3. **Conclusion**: This does not match the original expression.

### Final Conclusion
After evaluating all options, we find that only the first option \(2(x - 4)(x + 1)\) is equivalent to \(2x^2 - 6x - 8\).