1. **State the problem:** Divide the expression $$-18j^2 + 18j + 36$$ by $$6j$$ and express the result including any remainder as a simplified fraction. 2. **Write the division as a fraction:** $$\frac{-18j^2 + 18j + 36}{6j}$$ 3. **Split the fraction into separate terms:** $$\frac{-18j^2}{6j} + \frac{18j}{6j} + \frac{36}{6j}$$ 4. **Simplify each term:** - For $$\frac{-18j^2}{6j}$$, cancel common factors: $$\frac{\cancel{-18}j^{\cancel{2}}}{\cancel{6}j^{\cancel{1}}} = -3j$$ - For $$\frac{18j}{6j}$$, cancel common factors: $$\frac{\cancel{18}j}{\cancel{6}j} = 3$$ - For $$\frac{36}{6j}$$, simplify numerator and denominator: $$\frac{\cancel{36}}{\cancel{6}j} = \frac{6}{j}$$ 5. **Combine the simplified terms:** $$-3j + 3 + \frac{6}{j}$$ 6. **Final answer:** The quotient is $$-3j + 3$$ with a remainder expressed as the fraction $$\frac{6}{j}$$. Thus, $$\frac{-18j^2 + 18j + 36}{6j} = -3j + 3 + \frac{6}{j}$$