1. **State the problem:** Simplify the expression $$\frac{4-\sqrt{2}}{2\sqrt{8}}$$. 2. **Recall the formula and rules:** To simplify a fraction with radicals, simplify the radicals first and then reduce the fraction by canceling common factors. 3. **Simplify the denominator:** $$\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$$. 4. **Substitute back:** $$\frac{4-\sqrt{2}}{2 \times 2\sqrt{2}} = \frac{4-\sqrt{2}}{4\sqrt{2}}$$. 5. **Rewrite the fraction:** $$\frac{4-\sqrt{2}}{4\sqrt{2}} = \frac{4}{4\sqrt{2}} - \frac{\sqrt{2}}{4\sqrt{2}}$$. 6. **Simplify each term:** - $$\frac{4}{4\sqrt{2}} = \frac{\cancel{4}}{\cancel{4}\sqrt{2}} = \frac{1}{\sqrt{2}}$$ - $$\frac{\sqrt{2}}{4\sqrt{2}} = \frac{\cancel{\sqrt{2}}}{4\cancel{\sqrt{2}}} = \frac{1}{4}$$ 7. **So the expression becomes:** $$\frac{1}{\sqrt{2}} - \frac{1}{4}$$. 8. **Rationalize the first term:** Multiply numerator and denominator by $$\sqrt{2}$$: $$\frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$$. 9. **Rewrite the expression:** $$\frac{\sqrt{2}}{2} - \frac{1}{4}$$. 10. **Find common denominator 4:** $$\frac{\sqrt{2}}{2} = \frac{2\sqrt{2}}{4}$$. 11. **Subtract:** $$\frac{2\sqrt{2}}{4} - \frac{1}{4} = \frac{2\sqrt{2} - 1}{4}$$. **Final answer:** $$\boxed{\frac{2\sqrt{2} - 1}{4}}$$