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Choose the correct simplification of [tex]\((4x-3)(3x^2-4x-3)\)[/tex].

A. [tex]\(12x^3 + 25x^2 + 9\)[/tex]

B. [tex]\(12x^3 - 25x^2 - 9\)[/tex]

C. [tex]\(12x^3 + 25x^2 - 9\)[/tex]

D. [tex]\(12x^3 - 25x^2 + 9\)[/tex]

Answer :

Let's simplify the expression [tex]\((4x-3)(3x^2-4x-3)\)[/tex] step-by-step by using the distributive property, also known as the FOIL method for multiplying polynomials:

1. Multiply each term in the first polynomial by each term in the second polynomial:

- Multiply [tex]\(4x\)[/tex] by each term in the second polynomial:
[tex]\[
4x \cdot 3x^2 = 12x^3
\][/tex]
[tex]\[
4x \cdot (-4x) = -16x^2
\][/tex]
[tex]\[
4x \cdot (-3) = -12x
\][/tex]

- Multiply [tex]\(-3\)[/tex] by each term in the second polynomial:
[tex]\[
-3 \cdot 3x^2 = -9x^2
\][/tex]
[tex]\[
-3 \cdot (-4x) = 12x
\][/tex]
[tex]\[
-3 \cdot (-3) = 9
\][/tex]

2. Combine the results:

- Combine all the terms:
[tex]\[
12x^3 - 16x^2 - 12x - 9x^2 + 12x + 9
\][/tex]

3. Combine like terms:

- Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
-16x^2 - 9x^2 = -25x^2
\][/tex]

- Combine the [tex]\(x\)[/tex] terms:
[tex]\[
-12x + 12x = 0
\][/tex]

- Now, write the simplified expression:
[tex]\[
12x^3 - 25x^2 + 9
\][/tex]

So, the correct simplification of [tex]\((4x-3)(3x^2-4x-3)\)[/tex] is [tex]\(\boxed{12x^3 - 25x^2 + 9}\)[/tex].

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