1. **State the problem:** Evaluate $$\frac{3}{2}-\sqrt{3} - \left(\frac{2}{3}+\sqrt{3}\right)$$ without using a calculator and express the answer in the form $$m+n\sqrt{3}$$. 2. **Rewrite the expression:** $$\frac{3}{2} - \sqrt{3} - \frac{2}{3} - \sqrt{3}$$ 3. **Group like terms:** $$\left(\frac{3}{2} - \frac{2}{3}\right) + \left(-\sqrt{3} - \sqrt{3}\right)$$ 4. **Find common denominator and simplify the rational part:** $$\frac{3}{2} - \frac{2}{3} = \frac{3 \times 3}{2 \times 3} - \frac{2 \times 2}{3 \times 2} = \frac{9}{6} - \frac{4}{6} = \frac{9 - 4}{6} = \frac{5}{6}$$ 5. **Simplify the irrational part:** $$-\sqrt{3} - \sqrt{3} = -2\sqrt{3}$$ 6. **Combine the simplified parts:** $$\frac{5}{6} - 2\sqrt{3}$$ **Final answer:** $$\frac{5}{6} - 2\sqrt{3}$$ where $$m=\frac{5}{6}$$ and $$n=-2$$.