To find the product of [tex]\((a + 3)\)[/tex] and [tex]\((-2a^2 + 15a + 6b^2)\)[/tex], we need to use the distributive property (also known as the FOIL method for binomials) to multiply each term in the first expression by each term in the second expression.
Let's go step by step:
1. Multiply [tex]\(a\)[/tex] by each term in [tex]\((-2a^2 + 15a + 6b^2)\)[/tex]:
[tex]\[
a \cdot (-2a^2) = -2a^3
\][/tex]
[tex]\[
a \cdot 15a = 15a^2
\][/tex]
[tex]\[
a \cdot 6b^2 = 6ab^2
\][/tex]
2. Multiply [tex]\(3\)[/tex] by each term in [tex]\((-2a^2 + 15a + 6b^2)\)[/tex]:
[tex]\[
3 \cdot (-2a^2) = -6a^2
\][/tex]
[tex]\[
3 \cdot 15a = 45a
\][/tex]
[tex]\[
3 \cdot 6b^2 = 18b^2
\][/tex]
3. Now, add all the results together:
[tex]\[
-2a^3 + 15a^2 + 6ab^2 + (-6a^2) + 45a + 18b^2
\][/tex]
4. Combine like terms:
[tex]\[
-2a^3 + (15a^2 - 6a^2) + 6ab^2 + 45a + 18b^2
\][/tex]
[tex]\[
-2a^3 + 9a^2 + 6ab^2 + 45a + 18b^2
\][/tex]
Thus, the correct product of [tex]\((a + 3)\)[/tex] and [tex]\((-2a^2 + 15a + 6b^2)\)[/tex] is:
[tex]\[
-2a^3 + 9a^2 + 6ab^2 + 45a + 18b^2
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{-2a^3 + 9a^2 + 45a + 6ab^2 + 18b^2}
\][/tex]
