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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]

We start with the expression:
[tex]$$
(4x - 3)(3x^2 - 4x - 3).
$$[/tex]

Step 1. Distribute each term of the first factor into the second factor.

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

Next, multiply [tex]$-3$[/tex] by each term in the second factor:
[tex]$$
-3 \cdot 3x^2 = -9x^2,
$$[/tex]
[tex]$$
-3 \cdot (-4x) = +12x,
$$[/tex]
[tex]$$
-3 \cdot (-3) = +9.
$$[/tex]

Step 2. Combine all the terms.

Gather the terms calculated:
[tex]$$
12x^3 \quad (-16x^2 - 9x^2) \quad (-12x + 12x) \quad +9.
$$[/tex]

Combine like terms:
- For [tex]$x^3$[/tex]:
[tex]$$
12x^3.
$$[/tex]
- For [tex]$x^2$[/tex]:
[tex]$$
-16x^2 - 9x^2 = -25x^2.
$$[/tex]
- For [tex]$x$[/tex]:
[tex]$$
-12x + 12x = 0.
$$[/tex]
- The constant term is:
[tex]$$
+9.
$$[/tex]

Thus, the expression simplifies to:
[tex]$$
12x^3 - 25x^2 + 9.
$$[/tex]

Step 3. Identify the correct answer.

The simplified expression is:
[tex]$$
12x^3 - 25x^2 + 9.
$$[/tex]

This corresponds to the option:
[tex]$$
\textbf{12 x}^3 - 25 \textbf{ x}^2 + 9.
$$[/tex]