1. **State the problem:** Simplify the expression $$\frac{2x^2 - 8}{4x}$$. 2. **Formula and rules:** To simplify a rational expression, factor the numerator and denominator and then cancel common factors. 3. **Factor the numerator:** $$2x^2 - 8 = 2(x^2 - 4)$$. 4. **Recognize difference of squares:** $$x^2 - 4 = (x - 2)(x + 2)$$. 5. **Rewrite the expression:** $$\frac{2(x - 2)(x + 2)}{4x}$$. 6. **Factor the denominator:** $$4x = 2 \times 2x$$. 7. **Rewrite the expression with factored denominator:** $$\frac{2(x - 2)(x + 2)}{2 \times 2x}$$. 8. **Cancel common factor 2:** $$\frac{\cancel{2}(x - 2)(x + 2)}{\cancel{2} \times 2x} = \frac{(x - 2)(x + 2)}{2x}$$. 9. **Final simplified form:** $$\frac{(x - 2)(x + 2)}{2x}$$. This is the simplest form unless further expansion or specific values for $x$ are given.