1. **State the problem:** Simplify the expression $$\frac{2}{\sqrt{2} - 1}$$. 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. 3. **Multiply numerator and denominator by the conjugate:** $$\frac{2}{\sqrt{2} - 1} \times \frac{\sqrt{2} + 1}{\sqrt{2} + 1} = \frac{2(\sqrt{2} + 1)}{(\sqrt{2} - 1)(\sqrt{2} + 1)}$$ 4. **Simplify the denominator using difference of squares:** $$ (\sqrt{2})^2 - 1^2 = 2 - 1 = 1 $$ 5. **Substitute back:** $$ \frac{2(\sqrt{2} + 1)}{1} = 2\sqrt{2} + 2 $$ 6. **Final answer:** $$ 2\sqrt{2} + 2 $$ This is the simplified form of the original expression.