Identify two expressions that will be equivalent to 2x^2−8x−10

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

(1 point)
Option #___ and Option #___

GPT-4o mini · Answer

To find the expressions that are equivalent to $2 x^{2} - 8 x - 10$ when distributed, we should expand each option and compare them to see which ones match the original expression.

Option #1: $2 x (x - 4 x - 5)$
Distributing gives:
$2 x (x - 4 x - 5) = 2 x (- 3 x - 5) = - 6 x^{2} - 10 x (Not equivalent)$

Option #2: $(2 x + 2) (x - 5)$
Distributing gives:
$(2 x + 2) (x - 5) = 2 x \cdot x - 2 x \cdot 5 + 2 \cdot x - 2 \cdot 5 = 2 x^{2} - 10 x + 2 x - 10 = 2 x^{2} - 8 x - 10 (Equivalent)$

Option #3: $2 x (x - 5) + 2 (x - 5)$
Distributing gives:
$2 x (x - 5) + 2 (x - 5) = 2 x^{2} - 10 x + 2 x - 10 = 2 x^{2} - 8 x - 10 (Equivalent)$

Option #4: $(x - 5) (x - 3)$
Distributing gives:
$(x - 5) (x - 3) = x^{2} - 3 x - 5 x + 15 = x^{2} - 8 x + 15 (Not equivalent)$

Option #5: $- 6 x - 10$
This is a linear expression and does not match the original quadratic expression in any form.
$- 6 x - 10 (Not equivalent)$

The only options that are equivalent to the original expression $2 x^{2} - 8 x - 10$ are Option #2 and Option #3.

Therefore, the answer is:

Option #___ and Option #___
Option #2 and Option #3