Let's analyze each option to determine if it is a correct way to rewrite the expression [tex]\(\left(4 x^2+3 x-7\right)(x-2)\)[/tex] using the distributive property.
Option A:
[tex]\[
\left(4 x^2+3 x-7\right)(x) + \left(4 x^2+3 x-7\right)(x-2)
\][/tex]
This rewriting is incorrect because it doesn't properly apply the distributive property. The term [tex]\(\left(4 x^2+3 x-7\right)(x-2)\)[/tex] should be split into two separate products, [tex]\(\left(4 x^2+3 x-7\right)(x)\)[/tex] and [tex]\(\left(4 x^2+3 x-7\right)(-2)\)[/tex].
Option B:
[tex]\[
\left(4 x^2\right)(x-2) + (3 x)(x-2) + (-7)(x-2)
\][/tex]
This option correctly applies the distributive property by distributing each term in the polynomial [tex]\(4 x^2 + 3 x - 7\)[/tex] separately to the binomial [tex]\(x - 2\)[/tex].
Option C:
[tex]\[
\left(4 x^2\right)(x) + \left(4 x^2\right)(-2) + (3 x)(x) + (3 x)(-2) + (-7)(x) + (-7)(-2)
\][/tex]
This option correctly expands the expression by distributing each term individually. It fully breaks down the product into individual terms.
Option D:
[tex]\[
\left(4 x^2+3 x-7\right)(x) + \left(4 x^2+3 x-7\right)(-2)
\][/tex]
This option is another correct way to split the expression using the distributive property, where the binomial [tex]\(x - 2\)[/tex] is distributed across the polynomial [tex]\(4 x^2 + 3 x - 7\)[/tex].
From the analysis above, we see that Option A is the incorrect way to rewrite the expression. Therefore, the final conclusion is:
Option A is not a correct way to rewrite the expression using the distributive property.
