To determine the product of [tex]\((3x - 1)(4x^2 + 5)\)[/tex], follow these steps:
1. Distribute each term in the first expression [tex]\((3x - 1)\)[/tex] to each term in the second expression [tex]\((4x^2 + 5)\)[/tex]:
[tex]\[
(3x - 1)(4x^2 + 5)
\][/tex]
2. First, distribute [tex]\(3x\)[/tex] to each term in the second expression:
[tex]\[
3x \cdot 4x^2 + 3x \cdot 5
\][/tex]
- [tex]\(3x \cdot 4x^2 = 12x^3\)[/tex]
- [tex]\(3x \cdot 5 = 15x\)[/tex]
So, the result of the first part of the distribution is:
[tex]\[
12x^3 + 15x
\][/tex]
3. Second, distribute [tex]\(-1\)[/tex] to each term in the second expression:
[tex]\[
-1 \cdot 4x^2 + -1 \cdot 5
\][/tex]
- [tex]\(-1 \cdot 4x^2 = -4x^2\)[/tex]
- [tex]\(-1 \cdot 5 = -5\)[/tex]
So, the result of the second part of the distribution is:
[tex]\[
-4x^2 - 5
\][/tex]
4. Combine all the distributed parts together:
[tex]\[
12x^3 + 15x + (-4x^2) + (-5)
\][/tex]
5. Reorganize terms to write the final expression in standard form:
[tex]\[
12x^3 - 4x^2 + 15x - 5
\][/tex]
Thus, the product of [tex]\((3x - 1)(4x^2 + 5)\)[/tex] is:
[tex]\[
\boxed{12x^3 - 4x^2 + 15x - 5}
\][/tex]
So, the correct answer is \( \text{A} \.
