1. **State the problem:** Simplify the expression $$\frac{5}{2 - \sqrt{2}}$$. 2. **Formula and rule:** To simplify expressions with radicals in the denominator, multiply numerator and denominator by the conjugate of the denominator. The conjugate of $$2 - \sqrt{2}$$ is $$2 + \sqrt{2}$$. 3. **Multiply numerator and denominator by the conjugate:** $$\frac{5}{2 - \sqrt{2}} \times \frac{2 + \sqrt{2}}{2 + \sqrt{2}} = \frac{5(2 + \sqrt{2})}{(2 - \sqrt{2})(2 + \sqrt{2})}$$ 4. **Simplify the denominator using difference of squares:** $$(2 - \sqrt{2})(2 + \sqrt{2}) = 2^2 - (\sqrt{2})^2 = 4 - 2 = 2$$ 5. **Substitute back:** $$\frac{5(2 + \sqrt{2})}{2} = \frac{5 \times 2 + 5 \times \sqrt{2}}{2} = \frac{10 + 5\sqrt{2}}{2}$$ 6. **Split the fraction:** $$\frac{10}{2} + \frac{5\sqrt{2}}{2}$$ 7. **Simplify each term:** $$5 + \frac{5\sqrt{2}}{2}$$ **Final answer:** $$5 + \frac{5\sqrt{2}}{2}$$