Question

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

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

Answer 1

To identify which expressions are equivalent to \( 2x^2 - 8x - 10 \) when distributed, we will expand each option:

**Option #1: \( 2x(x - 4x - 5) \)**
Let's first clarify the expression:
\( 2x(x - 4x - 5) = 2x(-3x - 5) = -6x^2 - 10x \) (after distributing).
This is not equivalent to \( 2x^2 - 8x - 10 \).

**Option #2: \( (2x + 2)(x - 5) \)**
Distributing:
\( (2x + 2)(x - 5) = 2x^2 - 10x + 2x - 10 = 2x^2 - 8x - 10 \).
This is equivalent to \( 2x^2 - 8x - 10 \).

**Option #3: \( 2x(x - 5) + 2(x - 5) \)**
Factoring out \( (x - 5) \):
\( 2x(x - 5) + 2(x - 5) = (2x + 2)(x - 5) \).
We already determined that \( (2x + 2)(x - 5) \) is equivalent to \( 2x^2 - 8x - 10 \).
Thus, this is also equivalent to \( 2x^2 - 8x - 10 \).

**Option #4: \( (x - 5)(x - 3) \)**
Distributing:
\( (x - 5)(x - 3) = x^2 - 3x - 5x + 15 = x^2 - 8x + 15 \).
This is not equivalent to \( 2x^2 - 8x - 10 \).

**Option #5: \( -6x - 10 \)**
This is a linear expression, not a quadratic one, and cannot be equivalent since no term matches \( 2x^2 \).

The two options that are equivalent to \( 2x^2 - 8x - 10 \) are:
- Option #2: \( (2x + 2)(x - 5) \)
- Option #3: \( 2x(x - 5) + 2(x - 5) \)

Thus, the answer is:
**Option #2 and Option #3** (with Option #2 being the lesser numbered option).