1. **State the problem:** Simplify the expression $$\frac{\frac{3}{2} + \frac{3}{x}}{2 + \frac{4}{x}}$$. 2. **Rewrite the expression:** To simplify, first find a common denominator for the terms inside the numerator and denominator. 3. **Simplify numerator:** $$\frac{3}{2} + \frac{3}{x} = \frac{3x}{2x} + \frac{6}{2x} = \frac{3x + 6}{2x}$$ 4. **Simplify denominator:** $$2 + \frac{4}{x} = \frac{2x}{x} + \frac{4}{x} = \frac{2x + 4}{x}$$ 5. **Rewrite the original expression:** $$\frac{\frac{3x + 6}{2x}}{\frac{2x + 4}{x}} = \frac{3x + 6}{2x} \times \frac{x}{2x + 4}$$ 6. **Cancel common factors:** $$= \frac{3x + 6}{\cancel{2x}} \times \frac{\cancel{x}}{2x + 4} = \frac{3x + 6}{2} \times \frac{1}{2x + 4}$$ 7. **Factor numerator and denominator:** $$3x + 6 = 3(x + 2)$$ $$2x + 4 = 2(x + 2)$$ 8. **Substitute back:** $$\frac{3(x + 2)}{2} \times \frac{1}{2(x + 2)}$$ 9. **Cancel common factor $(x + 2)$:** $$= \frac{3\cancel{(x + 2)}}{2} \times \frac{1}{2\cancel{(x + 2)}} = \frac{3}{2} \times \frac{1}{2} = \frac{3}{4}$$ **Final answer:** $$\frac{3}{4}$$