1. **State the problem:** Simplify the expression $$\frac{-216 \left(e^{\frac{1}{2}}\right)^5}{\left(3 e^{\frac{1}{4}}\right)^2}$$ 2. **Simplify the powers of $e$ in the numerator:** $$\left(e^{\frac{1}{2}}\right)^5 = e^{\frac{1}{2} \times 5} = e^{\frac{5}{2}}$$ So the numerator becomes: $$-216 e^{\frac{5}{2}}$$ 3. **Simplify the denominator:** $$\left(3 e^{\frac{1}{4}}\right)^2 = 3^2 \times \left(e^{\frac{1}{4}}\right)^2 = 9 e^{\frac{1}{2}}$$ 4. **Rewrite the entire expression:** $$\frac{-216 e^{\frac{5}{2}}}{9 e^{\frac{1}{2}}}$$ 5. **Divide the coefficients:** $$\frac{-216}{9} = -24$$ 6. **Divide the powers of $e$ using the property $\frac{e^a}{e^b} = e^{a-b}$:** $$e^{\frac{5}{2} - \frac{1}{2}} = e^2$$ 7. **Combine the results:** $$-24 e^2$$ **Final answer:** $$\boxed{-24 e^2}$$