High School

Simplify the expression:

[tex]
\[
(x^2 - 5x + 6)(2x - 3) - (3x^2 + 4x - 5)(x - 2)
\]
[/tex]

A. [tex] -x^3 + 11x^2 + 40x + 28 [/tex]

B. [tex] x^4 - 11x^3 - 40x + 28 [/tex]

C. [tex] x^4 + 11x^3 - 40x - 28 [/tex]

D. [tex] -x^3 - 11x^2 + 40x - 28 [/tex]

Answer :

To simplify the expression [tex]\(\left(x^2 - 5x + 6\right)(2x - 3) - \left(3x^2 + 4x - 5\right)(x - 2)\)[/tex], we'll follow these steps:

1. Expand Each Part Separately:

- First, we expand [tex]\((x^2 - 5x + 6)(2x - 3)\)[/tex]:
[tex]\[
(x^2 - 5x + 6)(2x - 3) = x^2(2x - 3) - 5x(2x - 3) + 6(2x - 3)
\][/tex]

[tex]\[
= (2x^3 - 3x^2) - (10x^2 - 15x) + (12x - 18)
\][/tex]

[tex]\[
= 2x^3 - 3x^2 - 10x^2 + 15x + 12x - 18
\][/tex]

[tex]\[
= 2x^3 - 13x^2 + 27x - 18
\][/tex]

- Next, expand [tex]\((3x^2 + 4x - 5)(x - 2)\)[/tex]:
[tex]\[
(3x^2 + 4x - 5)(x - 2) = 3x^2(x - 2) + 4x(x - 2) - 5(x - 2)
\][/tex]

[tex]\[
= (3x^3 - 6x^2) + (4x^2 - 8x) - (5x - 10)
\][/tex]

[tex]\[
= 3x^3 - 6x^2 + 4x^2 - 8x - 5x + 10
\][/tex]

[tex]\[
= 3x^3 - 2x^2 - 13x + 10
\][/tex]

2. Subtract the Second Expansion from the First:

Now subtract the second expression from the first:
[tex]\[
(2x^3 - 13x^2 + 27x - 18) - (3x^3 - 2x^2 - 13x + 10)
\][/tex]

[tex]\[
= 2x^3 - 13x^2 + 27x - 18 - 3x^3 + 2x^2 + 13x - 10
\][/tex]

[tex]\[
= (2x^3 - 3x^3) + (-13x^2 + 2x^2) + (27x + 13x) + (-18 - 10)
\][/tex]

[tex]\[
= -x^3 - 11x^2 + 40x - 28
\][/tex]

3. Determine the Correct Option:

After simplifying, the expression becomes [tex]\(-x^3 - 11x^2 + 40x - 28\)[/tex].

Thus, the correct choice is:
- Option d: [tex]\(-x^3 - 11x^2 + 40x - 28\)[/tex]

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