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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]

Sure! Let's simplify the expression [tex]\((4x - 3)(3x^2 - 4x - 3)\)[/tex] step-by-step:

1. Distribute [tex]\((4x - 3)\)[/tex] across each term in [tex]\((3x^2 - 4x - 3)\)[/tex]. This means we'll multiply [tex]\(4x\)[/tex] by each term in the second polynomial, and then do the same with [tex]\(-3\)[/tex].

2. Multiply [tex]\(4x\)[/tex] by each of the terms in [tex]\((3x^2 - 4x - 3)\)[/tex]:

[tex]\[
4x \cdot 3x^2 = 12x^3
\][/tex]

[tex]\[
4x \cdot (-4x) = -16x^2
\][/tex]

[tex]\[
4x \cdot (-3) = -12x
\][/tex]

3. Multiply [tex]\(-3\)[/tex] by each of the terms in [tex]\((3x^2 - 4x - 3)\)[/tex]:

[tex]\[
-3 \cdot 3x^2 = -9x^2
\][/tex]

[tex]\[
-3 \cdot (-4x) = 12x
\][/tex]

[tex]\[
-3 \cdot (-3) = 9
\][/tex]

4. Combine all these terms together:

[tex]\[
12x^3 - 16x^2 - 12x - 9x^2 + 12x + 9
\][/tex]

5. 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]

6. Write the final simplified expression:

[tex]\[
12x^3 - 25x^2 + 9
\][/tex]

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