To find the product of [tex]\((3a + 2)(4a^2 - 2a + 9)\)[/tex], let's multiply each term in the first polynomial by each term in the second polynomial, and then combine like terms. Here's the step-by-step process:
1. Multiply [tex]\(3a\)[/tex] by each term in [tex]\(4a^2 - 2a + 9\)[/tex]:
- [tex]\(3a \cdot 4a^2 = 12a^3\)[/tex]
- [tex]\(3a \cdot (-2a) = -6a^2\)[/tex]
- [tex]\(3a \cdot 9 = 27a\)[/tex]
2. Multiply [tex]\(2\)[/tex] by each term in [tex]\(4a^2 - 2a + 9\)[/tex]:
- [tex]\(2 \cdot 4a^2 = 8a^2\)[/tex]
- [tex]\(2 \cdot (-2a) = -4a\)[/tex]
- [tex]\(2 \cdot 9 = 18\)[/tex]
3. Combine all these results:
- From the first set: [tex]\(12a^3, -6a^2, 27a\)[/tex]
- From the second set: [tex]\(8a^2, -4a, 18\)[/tex]
4. Add the like terms:
- For [tex]\(a^3\)[/tex]: [tex]\(12a^3\)[/tex]
- For [tex]\(a^2\)[/tex]: [tex]\(-6a^2 + 8a^2 = 2a^2\)[/tex]
- For [tex]\(a\)[/tex]: [tex]\(27a - 4a = 23a\)[/tex]
- For the constant term: [tex]\(18\)[/tex]
Putting it all together, the final result is:
[tex]\[ 12a^3 + 2a^2 + 23a + 18 \][/tex]
Among the given options, the correct answer is:
[tex]\[ \boxed{12a^3 + 2a^2 + 23a + 18} \][/tex]
