To find the product of the expressions [tex]\(a + 3\)[/tex] and [tex]\(-2a^2 + 15a + 6b^2\)[/tex], we need to distribute each term in the first expression [tex]\(a + 3\)[/tex] to each term in the second expression [tex]\(-2a^2 + 15a + 6b^2\)[/tex]. Let's work through this step-by-step:
1. Distribute [tex]\(a\)[/tex] to 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. Distribute [tex]\(3\)[/tex] to 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. Combine all the terms:
[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]
Therefore, the 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].
Among the given options, the correct product is:
[tex]\[
-2a^3 + 9a^2 + 45a + 6ab^2 + 18b^2
\][/tex]
