1. **State the problem:** Divide the expression $$5 n^3 + 4 n^2 + 2 n$$ by $$4 n$$. 2. **Write the division as a fraction:** $$\frac{5 n^3 + 4 n^2 + 2 n}{4 n}$$ 3. **Split the fraction into separate terms:** $$\frac{5 n^3}{4 n} + \frac{4 n^2}{4 n} + \frac{2 n}{4 n}$$ 4. **Simplify each term by canceling common factors:** - For the first term: $$\frac{5 n^3}{4 n} = \frac{5 \cancel{n^3}}{4 \cancel{n}} = \frac{5 n^{3-1}}{4} = \frac{5 n^2}{4}$$ - For the second term: $$\frac{4 n^2}{4 n} = \frac{\cancel{4} n^2}{\cancel{4} n} = n^{2-1} = n$$ - For the third term: $$\frac{2 n}{4 n} = \frac{2 \cancel{n}}{4 \cancel{n}} = \frac{2}{4} = \frac{1}{2}$$ 5. **Combine the simplified terms:** $$\frac{5 n^2}{4} + n + \frac{1}{2}$$ 6. **Compare with the options:** This matches Option 1. **Final answer:** Option 1: $$\frac{5 n^2}{4} + n + \frac{1}{2}$$