1. **Problem Statement:** Mercedes multiplied $4x^3$ by the polynomial $(x^3 + 2x^2 - 3)$ and got $4x^9 + 8x^6 - 12x^3$. We need to find her error. 2. **Formula and Rules:** When multiplying a monomial by a polynomial, distribute the monomial to each term of the polynomial: $$a(b + c + d) = ab + ac + ad$$ Also, when multiplying powers with the same base, add the exponents: $$x^m \cdot x^n = x^{m+n}$$ 3. **Step-by-step Multiplication:** Multiply $4x^3$ by each term: - $4x^3 \cdot x^3 = 4x^{3+3} = 4x^6$ - $4x^3 \cdot 2x^2 = 8x^{3+2} = 8x^5$ - $4x^3 \cdot (-3) = -12x^3$ 4. **Intermediate Expression:** $$4x^6 + 8x^5 - 12x^3$$ 5. **Compare with Mercedes's result:** Mercedes wrote $4x^9 + 8x^6 - 12x^3$. 6. **Error Explanation:** Mercedes incorrectly added exponents when multiplying $4x^3$ by $x^3$ and $2x^2$. Instead of adding $3 + 3 = 6$ and $3 + 2 = 5$, she wrote $9$ and $6$ respectively, which is incorrect. **Final answer:** Mercedes's error was in the exponent addition during multiplication; she incorrectly calculated $x^{3+3}$ as $x^9$ and $x^{3+2}$ as $x^6$ instead of $x^6$ and $x^5$ respectively.