Question: What is the product of $5x^3$ and $xy^4 - 2x^3y$?
1. **State the problem:** We need to find the product of $5x^3$ and the expression $xy^4 - 2x^3y$.
2. **Write the expression:**
$$5x^3 \times (xy^4 - 2x^3y)$$
3. **Use the distributive property:** Multiply $5x^3$ by each term inside the parentheses:
$$5x^3 \times xy^4 - 5x^3 \times 2x^3y$$
4. **Multiply the coefficients and variables:**
- For the first term: $5 \times 1 = 5$ and $x^3 \times x = x^{3+1} = x^4$, so the term is $5x^4y^4$.
- For the second term: $5 \times 2 = 10$ and $x^3 \times x^3 = x^{3+3} = x^6$, so the term is $10x^6y$.
5. **Combine the terms with subtraction:**
$$5x^4y^4 - 10x^6y$$
6. **Check the options:** The correct product matches the option:
$5x^4y^4 - 10x^6y$.
**Final answer:** $5x^4y^4 - 10x^6y$