1. **State the problem:** Simplify the expression $$\frac{3}{4}x^2 - \frac{1}{2}y^2 - \frac{2}{5}xy + \frac{1}{6}y^2 + \frac{1}{10}xy + \frac{1}{3}y^2.$$ 2. **Combine like terms:** Group terms with $x^2$, $y^2$, and $xy$ separately. - $x^2$ terms: $$\frac{3}{4}x^2$$ - $y^2$ terms: $$-\frac{1}{2}y^2 + \frac{1}{6}y^2 + \frac{1}{3}y^2$$ - $xy$ terms: $$-\frac{2}{5}xy + \frac{1}{10}xy$$ 3. **Simplify $y^2$ terms:** Find common denominator for $$-\frac{1}{2}, \frac{1}{6}, \frac{1}{3}$$ which is 6. $$-\frac{1}{2} = -\frac{3}{6}, \quad \frac{1}{6} = \frac{1}{6}, \quad \frac{1}{3} = \frac{2}{6}$$ Sum: $$-\frac{3}{6} + \frac{1}{6} + \frac{2}{6} = \frac{0}{6} = 0$$ 4. **Simplify $xy$ terms:** Find common denominator for $$-\frac{2}{5}, \frac{1}{10}$$ which is 10. $$-\frac{2}{5} = -\frac{4}{10}, \quad \frac{1}{10} = \frac{1}{10}$$ Sum: $$-\frac{4}{10} + \frac{1}{10} = -\frac{3}{10}$$ 5. **Write the simplified expression:** $$\frac{3}{4}x^2 + 0 \cdot y^2 - \frac{3}{10}xy = \frac{3}{4}x^2 - \frac{3}{10}xy$$ **Final answer:** $$\boxed{\frac{3}{4}x^2 - \frac{3}{10}xy}$$