To determine which option is not a correct way to rewrite the given expression [tex]\((3x^2 + 4x - 7)(x - 2)\)[/tex] using the distributive property, let's analyze each provided option in detail.
### Option A:
[tex]\[
(3x^2)(x) + (3x^2)(-2) + (4x)(x) + (4x)(-2) + (-7)(x) + (-7)(-2)
\][/tex]
Here, the distributive property is applied correctly by individually multiplying each term of [tex]\((3x^2 + 4x - 7)\)[/tex] by each term in [tex]\((x - 2)\)[/tex]. This is a proper application of the distributive property.
### Option B:
[tex]\[
(3x^2)(x - 2) + (4x)(x - 2) + (-7)(x - 2)
\][/tex]
This option groups terms and applies the distributive property to the polynomial [tex]\((3x^2 + 4x - 7)\)[/tex] as a sum. Each term from the first polynomial is multiplied by the binomial [tex]\((x - 2)\)[/tex], which is also correct.
### Option C:
[tex]\[
(3x^2 + 4x - 7)(x) + (3x^2 + 4x - 7)(x - 2)
\][/tex]
In this option, the expression [tex]\((3x^2 + 4x - 7)\)[/tex] is multiplicatively duplicated with the terms [tex]\(x\)[/tex] and [tex]\((x - 2)\)[/tex]. This does not correctly apply the distributive property, as it wrongfully suggests a separation and repetition of terms, which is not mathematically correct.
### Option D:
[tex]\[
(3x^2 + 4x - 7)(x) + (3x^2 + 4x - 7)(-2)
\][/tex]
Similar to Option C, this option incorrectly splits [tex]\((3x^2 + 4x - 7)\)[/tex] and applies it multiplicatively with both [tex]\(x\)[/tex] and [tex]\(-2\)[/tex] separately, which follows an improper form for applying the distributive property.
Therefore, the incorrect way to rewrite the expression [tex]\((3 x^2 + 4 x - 7)(x - 2)\)[/tex] using the distributive property is:
[tex]\[
\boxed{\text{C}}
\][/tex]
Option C is not a correct application of the distributive property.
