Answer :
Let's analyze the expression [tex]\((2x^2 + 4x - 7)(x - 2)\)[/tex] and identify which one of the given options does not correctly use the distributive property.
### Option A:
[tex]\[ (2x^2 + 4x - 7)(x) + (2x^2 + 4x - 7)(-2) \][/tex]
This correctly applies the distributive property by splitting [tex]\((x - 2)\)[/tex] into [tex]\(x\)[/tex] and [tex]\(-2\)[/tex] and distributing [tex]\((2x^2 + 4x - 7)\)[/tex] across both terms:
[tex]\[ (2x^2 + 4x - 7)x + (2x^2 + 4x - 7)(-2) \][/tex]
### Option B:
[tex]\[ (2x^2)(x) + (2x^2)(-2) + (4x)(x) + (4x)(-2) + (-7)(x) + (-7)(-2) \][/tex]
This also correctly applies the distributive property by distributing each term in [tex]\((2x^2 + 4x - 7)\)[/tex] across each term in [tex]\((x - 2)\)[/tex]:
[tex]\[ (2x^2)x + (2x^2)(-2) + (4x)x + (4x)(-2) + (-7)x + (-7)(-2) \][/tex]
### Option C:
[tex]\[ (2x^2 + 4 - 7)(x) + (2x^2 + 4x - 7)(x - 2) \][/tex]
This is problematic. First, notice that [tex]\(2x^2 + 4 - 7\)[/tex] is not equivalent to [tex]\(2x^2 + 4x - 7\)[/tex] because the [tex]\(4x\)[/tex] term is missing from the first part of the expression on the left. Therefore, it does not correctly rewrite the original expression using the distributive property.
### Option D:
[tex]\[ (2x^2)(x - 2) + (4x)(x - 2) + (-7)(x - 2) \][/tex]
This correctly applies the distributive property by splitting [tex]\((2x^2 + 4x - 7)\)[/tex] and distributing each term across [tex]\((x - 2)\)[/tex]:
[tex]\[ (2x^2)(x - 2) + (4x)(x - 2) + (-7)(x - 2) \][/tex]
### Conclusion:
The correct answer is that Option C does not correctly use the distributive property:
[tex]\[ (2x^2 + 4 - 7)(x) + (2x^2 + 4x - 7)(x - 2) \][/tex]
Thus, the incorrect option is:
[tex]\[ \boxed{C} \][/tex]
### Option A:
[tex]\[ (2x^2 + 4x - 7)(x) + (2x^2 + 4x - 7)(-2) \][/tex]
This correctly applies the distributive property by splitting [tex]\((x - 2)\)[/tex] into [tex]\(x\)[/tex] and [tex]\(-2\)[/tex] and distributing [tex]\((2x^2 + 4x - 7)\)[/tex] across both terms:
[tex]\[ (2x^2 + 4x - 7)x + (2x^2 + 4x - 7)(-2) \][/tex]
### Option B:
[tex]\[ (2x^2)(x) + (2x^2)(-2) + (4x)(x) + (4x)(-2) + (-7)(x) + (-7)(-2) \][/tex]
This also correctly applies the distributive property by distributing each term in [tex]\((2x^2 + 4x - 7)\)[/tex] across each term in [tex]\((x - 2)\)[/tex]:
[tex]\[ (2x^2)x + (2x^2)(-2) + (4x)x + (4x)(-2) + (-7)x + (-7)(-2) \][/tex]
### Option C:
[tex]\[ (2x^2 + 4 - 7)(x) + (2x^2 + 4x - 7)(x - 2) \][/tex]
This is problematic. First, notice that [tex]\(2x^2 + 4 - 7\)[/tex] is not equivalent to [tex]\(2x^2 + 4x - 7\)[/tex] because the [tex]\(4x\)[/tex] term is missing from the first part of the expression on the left. Therefore, it does not correctly rewrite the original expression using the distributive property.
### Option D:
[tex]\[ (2x^2)(x - 2) + (4x)(x - 2) + (-7)(x - 2) \][/tex]
This correctly applies the distributive property by splitting [tex]\((2x^2 + 4x - 7)\)[/tex] and distributing each term across [tex]\((x - 2)\)[/tex]:
[tex]\[ (2x^2)(x - 2) + (4x)(x - 2) + (-7)(x - 2) \][/tex]
### Conclusion:
The correct answer is that Option C does not correctly use the distributive property:
[tex]\[ (2x^2 + 4 - 7)(x) + (2x^2 + 4x - 7)(x - 2) \][/tex]
Thus, the incorrect option is:
[tex]\[ \boxed{C} \][/tex]
