1. **State the problem:** Simplify the expression $$\frac{6}{4 - \sqrt{5}}$$. 2. **Formula and rule:** To simplify a fraction with a radical in the denominator, multiply numerator and denominator by the conjugate of the denominator. The conjugate of $$4 - \sqrt{5}$$ is $$4 + \sqrt{5}$$. 3. **Multiply numerator and denominator by the conjugate:** $$\frac{6}{4 - \sqrt{5}} \times \frac{4 + \sqrt{5}}{4 + \sqrt{5}} = \frac{6(4 + \sqrt{5})}{(4 - \sqrt{5})(4 + \sqrt{5})}$$ 4. **Simplify the denominator using difference of squares:** $$(4 - \sqrt{5})(4 + \sqrt{5}) = 4^2 - (\sqrt{5})^2 = 16 - 5 = 11$$ 5. **Expand the numerator:** $$6(4 + \sqrt{5}) = 24 + 6\sqrt{5}$$ 6. **Write the simplified expression:** $$\frac{24 + 6\sqrt{5}}{11}$$ 7. **Final answer:** $$\boxed{\frac{24 + 6\sqrt{5}}{11}}$$