1. **State the problem:** Simplify the equation $$-rac{\frac{e^{xy} + x}{y} + \frac{x + 6xe^{x}}{y}}{\frac{y}{x} - y} = \frac{\log(xy) - \rho(x,y)}{\frac{h}{x} - \frac{h}{y}}$$ 2. **Combine terms in the numerator of the left fraction:** Since both terms share denominator $y$, add the numerators: $$\frac{e^{xy} + x}{y} + \frac{x + 6xe^{x}}{y} = \frac{e^{xy} + x + x + 6xe^{x}}{y} = \frac{e^{xy} + 2x + 6xe^{x}}{y}$$ 3. **Rewrite the left fraction:** $$-\frac{\frac{e^{xy} + 2x + 6xe^{x}}{y}}{\frac{y}{x} - y} = -\frac{e^{xy} + 2x + 6xe^{x}}{y \left(\frac{y}{x} - y\right)}$$ 4. **Simplify the denominator of the left fraction:** $$\frac{y}{x} - y = y\left(\frac{1}{x} - 1\right) = y \frac{1 - x}{x}$$ 5. **Substitute back:** $$-\frac{e^{xy} + 2x + 6xe^{x}}{y \cdot y \frac{1 - x}{x}} = -\frac{e^{xy} + 2x + 6xe^{x}}{y^{2} \frac{1 - x}{x}}$$ 6. **Rewrite the denominator to simplify division:** $$-\frac{e^{xy} + 2x + 6xe^{x}}{y^{2} \frac{1 - x}{x}} = -\frac{e^{xy} + 2x + 6xe^{x}}{y^{2}} \cdot \frac{x}{1 - x}$$ 7. **Final simplified left side:** $$-\frac{x \left(e^{xy} + 2x + 6xe^{x}\right)}{y^{2} (1 - x)}$$ 8. **Simplify the right fraction denominator:** $$\frac{h}{x} - \frac{h}{y} = h \left(\frac{1}{x} - \frac{1}{y}\right) = h \frac{y - x}{xy}$$ 9. **Rewrite the right fraction:** $$\frac{\log(xy) - \rho(x,y)}{h \frac{y - x}{xy}} = \frac{\log(xy) - \rho(x,y)}{h} \cdot \frac{xy}{y - x}$$ 10. **Rewrite denominator $y - x$ as $-(x - y)$ to match left side denominator:** $$\frac{xy}{y - x} = -\frac{xy}{x - y}$$ 11. **Final simplified right side:** $$-\frac{xy (\log(xy) - \rho(x,y))}{h (x - y)}$$ **Summary:** $$-\frac{x \left(e^{xy} + 2x + 6xe^{x}\right)}{y^{2} (1 - x)} = -\frac{xy (\log(xy) - \rho(x,y))}{h (x - y)}$$ This is the simplified form of the original equation.