Question

Identify two expressions that will be equivalent to 2x2−8x−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 determine which expressions are equivalent to \(2x^2 - 8x - 10\) when distributed, we need to simplify each option.

1. **Option #1: \(2x(x - 4x - 5)\)**

First, let's clarify the expression. It seems there is a typo or misplacement in the terms. Assuming it should be \(2x(x - 4) - 5\):

\[
2x(x - 4) - 5 = 2x^2 - 8x - 5
\]

This expression does not equal \(2x^2 - 8x - 10\).

2. **Option #2: \((2x + 2)(x - 5)\)**

Distributing gives:

\[
(2x + 2)(x - 5) = 2x^2 - 10x + 2x - 10 = 2x^2 - 8x - 10
\]

This expression equals \(2x^2 - 8x - 10\).

3. **Option #3: \(2x(x - 5) + 2(x - 5)\)**

First, factor \( (x - 5) \):

\[
= (2x + 2)(x - 5) = 2x^2 - 10x + 2x - 10 = 2x^2 - 8x - 10
\]

This expression equals \(2x^2 - 8x - 10\).

4. **Option #4: \((x - 5)(x - 3)\)**

Distributing gives:

\[
(x - 5)(x - 3) = x^2 - 3x - 5x + 15 = x^2 - 8x + 15
\]

This expression does not equal \(2x^2 - 8x - 10\).

5. **Option #5: \(-6x - 10\)**

This is simply a linear expression and does not equal \(2x^2 - 8x - 10\).

### Conclusion:
The equivalent expressions are:

- Option #2: \((2x + 2)(x - 5)\)
- Option #3: \(2x(x - 5) + 2(x - 5)\)

Given the requirement to enter the option with the lesser number first, the answer is:

**Option #2** and **Option #3**.