To find the product of [tex]\((a+3)\)[/tex] and [tex]\((-2a^2 + 15a + 6b^2)\)[/tex], we will perform polynomial multiplication by distributing each term in the first polynomial across the second polynomial.
Given:
[tex]\[ (a + 3) \][/tex]
[tex]\[ (-2a^2 + 15a + 6b^2) \][/tex]
First, distribute [tex]\(a\)[/tex]:
[tex]\[ a \cdot (-2a^2 + 15a + 6b^2) = -2a^3 + 15a^2 + 6ab^2 \][/tex]
Next, distribute [tex]\(3\)[/tex]:
[tex]\[ 3 \cdot (-2a^2 + 15a + 6b^2) = -6a^2 + 45a + 18b^2 \][/tex]
Now, combine the results from both distributions:
[tex]\[ -2a^3 + 15a^2 + 6ab^2 + (-6a^2) + 45a + 18b^2 \][/tex]
Combine like terms:
- The [tex]\(a^3\)[/tex] term: [tex]\(-2a^3\)[/tex]
- The [tex]\(a^2\)[/tex] terms: [tex]\(15a^2 - 6a^2 = 9a^2\)[/tex]
- The [tex]\(ab^2\)[/tex] term: [tex]\(6ab^2\)[/tex]
- The [tex]\(a\)[/tex] term: [tex]\(45a\)[/tex]
- The [tex]\(b^2\)[/tex] term: [tex]\(18b^2\)[/tex]
Thus, the product of [tex]\((a + 3)\)[/tex] and [tex]\((-2a^2 + 15a + 6b^2)\)[/tex] simplifies to:
[tex]\[ -2a^3 + 9a^2 + 6ab^2 + 45a + 18b^2 \][/tex]
Therefore, the correct answer is:
[tex]\[ -2a^3 + 9a^2 + 6a b^2 + 45a + 18b^2 \][/tex]
This matches the third option in the given choices:
[tex]\[ \boxed{-2 a^3 + 9 a^2 + 45 a + 6 a b^2 + 18 b^2} \][/tex]
