Identify two expressions that will be equivalent to 2x2−8x−10 when distributed. Enter the option with the lesser number first.

Question

Option #1: 2x(x−4x−5)
Option #2: (2x+2)(x−5)
Option #3: 2x(x−5)+2(x−5)
Option #4: (x−5)(x−3)
Option #5: −6x−10 (1 point)

Option # and Option #

Answer 1

To identify the expressions that are equivalent to \( 2x^2 - 8x - 10 \), we will explore the given options:

1. Option #1: \( 2x(x - 4x - 5) \) This expression seems incorrectly written; if it were \( 2x(x - 4) - 5 \), we would analyze it, but let's leave this expression for now.

2. Option #2: \( (2x + 2)(x - 5) \) Applying distribution: \[ (2x + 2)(x - 5) = 2x \cdot x + 2 \cdot x - 2x \cdot 5 - 2 \cdot 5 = 2x^2 - 10x + 2x - 10 = 2x^2 - 8x - 10 \] This expression is equivalent.

3. Option #3: \( 2x(x - 5) + 2(x - 5) \) This can also be factored: \[ 2x(x - 5) + 2(x - 5) = (2x + 2)(x - 5) \] We already showed that \( (2x + 2)(x - 5) \) is equivalent to \( 2x^2 - 8x - 10 \), so this option is also equivalent.

4. Option #4: \( (x - 5)(x - 3) \) Applying distribution: \[ (x - 5)(x - 3) = x^2 - 3x - 5x + 15 = x^2 - 8x + 15 \] This is not equivalent.

5. Option #5: \( -6x - 10 \) This expression is not equivalent since it lacks the \( x^2 \) term.

From this analysis, the two options that are equivalent to \( 2x^2 - 8x - 10 \) are Option #2 and Option #3.

Thus, the answer is: Option #2 and Option #3.