Answer :
To determine which option is not a correct way to rewrite the expression [tex]\(\left(2 x^2 + 4 x - 7 \right)(x - 2)\)[/tex] using the distributive property, let's closely examine each option step-by-step:
A. [tex]\(\left(2 x^2\right)(x) + \left(2 x^2\right)(-2) + (4 x)(x) + (4 x)(-2) + (-7)(x) + (-7)(-2)\)[/tex]
This option is correctly using the distributive property by distributing each term of [tex]\(\left(2 x^2 + 4 x - 7 \right)\)[/tex] to both [tex]\(x\)[/tex] and [tex]\(-2\)[/tex]. Thus, Option A is a correct rewriting.
B. [tex]\(\left(2 x^2 + 4 x - 7 \right)(x) + \left(2 x^2 + 4 x - 7 \right)(-2)\)[/tex]
This correctly shows the distribution of the entire polynomial [tex]\( \left(2 x^2 + 4 x - 7 \right) \)[/tex] to both [tex]\(x\)[/tex] and [tex]\(-2\)[/tex]. Thus, Option B is a correct rewriting.
C. [tex]\(\left(2 x^2\right)(x - 2) + (4 x)(x - 2) + (-7)(x - 2)\)[/tex]
This option is grouping [tex]\(\left(2 x^2 + 4 x - 7 \right)\)[/tex] and then distributing [tex]\(x - 2\)[/tex]. Hence, Option C is a correct rewriting.
D. [tex]\(\left(2 x^2 + 4 - 7 \right)(x) + \left(2 x^2 + 4 x - 7 \right)(x - 2)\)[/tex]
Notice in Option D, there is an error within the expression inside the parentheses. It should be [tex]\(\left(2 x^2 + 4 x - 7 \right)\)[/tex], but instead it is written as [tex]\(\left(2 x^2 + 4 - 7 \right)\)[/tex], which simplifies incorrectly. Therefore, Option D is not a correct rewriting.
So, the incorrect way to rewrite the expression using the distributive property is:
Option D. [tex]\(\left(2 x^2 + 4 - 7 \right)(x) + \left(2 x^2 + 4 x - 7 \right)(x - 2)\)[/tex]
A. [tex]\(\left(2 x^2\right)(x) + \left(2 x^2\right)(-2) + (4 x)(x) + (4 x)(-2) + (-7)(x) + (-7)(-2)\)[/tex]
This option is correctly using the distributive property by distributing each term of [tex]\(\left(2 x^2 + 4 x - 7 \right)\)[/tex] to both [tex]\(x\)[/tex] and [tex]\(-2\)[/tex]. Thus, Option A is a correct rewriting.
B. [tex]\(\left(2 x^2 + 4 x - 7 \right)(x) + \left(2 x^2 + 4 x - 7 \right)(-2)\)[/tex]
This correctly shows the distribution of the entire polynomial [tex]\( \left(2 x^2 + 4 x - 7 \right) \)[/tex] to both [tex]\(x\)[/tex] and [tex]\(-2\)[/tex]. Thus, Option B is a correct rewriting.
C. [tex]\(\left(2 x^2\right)(x - 2) + (4 x)(x - 2) + (-7)(x - 2)\)[/tex]
This option is grouping [tex]\(\left(2 x^2 + 4 x - 7 \right)\)[/tex] and then distributing [tex]\(x - 2\)[/tex]. Hence, Option C is a correct rewriting.
D. [tex]\(\left(2 x^2 + 4 - 7 \right)(x) + \left(2 x^2 + 4 x - 7 \right)(x - 2)\)[/tex]
Notice in Option D, there is an error within the expression inside the parentheses. It should be [tex]\(\left(2 x^2 + 4 x - 7 \right)\)[/tex], but instead it is written as [tex]\(\left(2 x^2 + 4 - 7 \right)\)[/tex], which simplifies incorrectly. Therefore, Option D is not a correct rewriting.
So, the incorrect way to rewrite the expression using the distributive property is:
Option D. [tex]\(\left(2 x^2 + 4 - 7 \right)(x) + \left(2 x^2 + 4 x - 7 \right)(x - 2)\)[/tex]
