1. **State the problem:** Simplify the expression $$\frac{-3x^{\frac{1}{2}}}{2x^{-\frac{3}{4}}}$$. 2. **Recall the rule for dividing powers with the same base:** $$\frac{a^m}{a^n} = a^{m-n}$$. 3. **Apply the rule to the powers of $x$:** $$\frac{x^{\frac{1}{2}}}{x^{-\frac{3}{4}}} = x^{\frac{1}{2} - (-\frac{3}{4})} = x^{\frac{1}{2} + \frac{3}{4}}$$ 4. **Calculate the exponent:** $$\frac{1}{2} + \frac{3}{4} = \frac{2}{4} + \frac{3}{4} = \frac{5}{4}$$ 5. **Rewrite the expression:** $$\frac{-3}{2} x^{\frac{5}{4}}$$ 6. **Final simplified expression:** $$-\frac{3}{2} x^{\frac{5}{4}}$$ 7. **Note:** The bottom-left term $2x^4$ is not part of the fraction to simplify here, so it is not included in the simplification.