1. **State the problem:** Divide the expression $4w^3 + 10w^2$ by $2w^2$ and express the result including any remainder as a simplified fraction. 2. **Write the division as a fraction:** $$\frac{4w^3 + 10w^2}{2w^2}$$ 3. **Split the fraction into two parts:** $$\frac{4w^3}{2w^2} + \frac{10w^2}{2w^2}$$ 4. **Simplify each term separately:** - For the first term: $$\frac{4w^3}{2w^2} = \frac{\cancel{4}^2 \times w^{\cancel{3}^1}}{\cancel{2}^1 \times w^{\cancel{2}^0}} = 2w$$ - For the second term: $$\frac{10w^2}{2w^2} = \frac{\cancel{10}^5 \times w^{\cancel{2}^0}}{\cancel{2}^1 \times w^{\cancel{2}^0}} = 5$$ 5. **Combine the simplified terms:** $$2w + 5$$ 6. **Conclusion:** The division results in $2w + 5$ with no remainder. **Final answer:** $2w + 5$