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