To find the product of $(a + 3)$ and $(-2a^2 + 15a + 6b^2)$, we'll use the distributive property (also known as the FOIL method for binomials).

Step-by-step expansion:

1. Multiply $a$ by each term in the second expression: $a \cdot (-2a^2) = -2a^3$ $a \cdot (15a) = 15a^2$ $a \cdot (6b^2) = 6ab^2$

2. Multiply $3$ by each term in the second expression: $3 \cdot (-2a^2) = -6a^2$ $3 \cdot (15a) = 45a$ $3 \cdot (6b^2) = 18b^2$

Now, combine all the terms: $-2a^3 + 15a^2 + 6ab^2 - 6a^2 + 45a + 18b^2$

Simplify by combining like terms: $-2a^3 + (15a^2 - 6a^2) + 6ab^2 + 45a + 18b^2$ $= -2a^3 + 9a^2 + 6ab^2 + 45a + 18b^2$

Thus, the expanded and simplified expression is: $-2a^3 + 9a^2 + 6ab^2 + 45a + 18b^2$

This matches the fourth option: $\boxed{-2a^3 + 9a^2 + 45a + 6ab^2 + 18b^2}$

---

Would you like more details on any step or have questions?

Here are five related questions to explore further:

1. What happens when you multiply a trinomial by a binomial?
2. How does the distributive property apply to expressions with multiple variables?
3. What if we change the signs of the terms in one of the factors?
4. Can we factor the resulting expression after expansion?
5. How does this product change if one term is raised to a higher power?

Tip: When expanding polynomials, always organize your terms carefully and combine like terms at the end.