To find the product of the two expressions [tex]\( (a + 3) \)[/tex] and [tex]\( -2a^2 + 15a + 6b^2 \)[/tex], we can use the distributive property (also known as the FOIL method for polynomials). Let's expand the product step-by-step.
Given the expressions:
[tex]\[
(a + 3)
\][/tex]
and
[tex]\[
-2a^2 + 15a + 6b^2
\][/tex]
### 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]
So, distributing [tex]\( a \)[/tex] yields:
[tex]\[ -2a^3 + 15a^2 + 6ab^2 \][/tex]
### Step 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]
So, distributing [tex]\( 3 \)[/tex] yields:
[tex]\[ -6a^2 + 45a + 18b^2 \][/tex]
### Step 3: Add the two results together:
Combining all the terms from the distribution, we get:
[tex]\[ (-2a^3 + 15a^2 + 6ab^2) + (-6a^2 + 45a + 18b^2) \][/tex]
Group and combine like terms:
- The [tex]\( a^3 \)[/tex] term: [tex]\(-2a^3\)[/tex]
- The [tex]\( a^2 \)[/tex] term: [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 expanded form of the product is:
[tex]\[ -2a^3 + 9a^2 + 6ab^2 + 45a + 18b^2 \][/tex]
So, the final expression is:
[tex]\[ -2a^3 + 9a^2 + 6ab^2 + 45a + 18b^2 \][/tex]
### Conclusion:
Comparing this final expression to the given options, we find that the correct answer is:
[tex]\[ \boxed{-2a^3 + 9a^2 + 45a + 6ab^2 + 18b^2} \][/tex]
