1. **State the problem:** Find the product of the polynomials $a+3$ and $-2a^2+15a+6b^2$. 2. **Formula and rules:** To multiply polynomials, use the distributive property: multiply each term in the first polynomial by each term in the second polynomial, then combine like terms. 3. **Multiply each term:** $$ (a+3)(-2a^2+15a+6b^2) = a(-2a^2+15a+6b^2) + 3(-2a^2+15a+6b^2) $$ 4. **Distribute:** $$ = a \cdot (-2a^2) + a \cdot 15a + a \cdot 6b^2 + 3 \cdot (-2a^2) + 3 \cdot 15a + 3 \cdot 6b^2 $$ $$ = -2a^3 + 15a^2 + 6ab^2 - 6a^2 + 45a + 18b^2 $$ 5. **Combine like terms:** $$ 15a^2 - 6a^2 = \cancel{15a^2} - \cancel{6a^2} = 9a^2 $$ 6. **Final simplified expression:** $$ -2a^3 + 9a^2 + 6ab^2 + 45a + 18b^2 $$ This matches none of the options exactly, but the closest is: -2a^3 + 21a^2 + 45a + 6ab^2 + 18b^2 (which has 21a^2 instead of 9a^2) Therefore, the correct product is: $$ \boxed{-2a^3 + 9a^2 + 6ab^2 + 45a + 18b^2} $$