1. **State the problem:** Simplify the expression $4x^2 y^3 \times 2x^3 y^4$. 2. **Recall the multiplication rules for exponents:** When multiplying terms with the same base, add the exponents: $a^m \times a^n = a^{m+n}$. 3. **Multiply the coefficients:** $4 \times 2 = 8$. 4. **Multiply the $x$ terms:** $x^2 \times x^3 = x^{2+3} = x^5$. 5. **Multiply the $y$ terms:** $y^3 \times y^4 = y^{3+4} = y^7$. 6. **Combine all parts:** The simplified expression is $$8x^5 y^7$$. This means you multiply the numbers and add the exponents of like variables.