1. **State the problem:** Simplify the expression $$\frac{2x^2 - 8}{x^2 - 4x} \div \frac{x - 4}{x}$$. 2. **Rewrite division as multiplication by reciprocal:** $$\frac{2x^2 - 8}{x^2 - 4x} \times \frac{x}{x - 4}$$ 3. **Factor all polynomials:** - Numerator of first fraction: $$2x^2 - 8 = 2(x^2 - 4) = 2(x - 2)(x + 2)$$ - Denominator of first fraction: $$x^2 - 4x = x(x - 4)$$ 4. **Substitute factored forms:** $$\frac{2(x - 2)(x + 2)}{x(x - 4)} \times \frac{x}{x - 4}$$ 5. **Multiply the fractions:** $$\frac{2(x - 2)(x + 2)}{x(x - 4)} \times \frac{x}{x - 4} = \frac{2(x - 2)(x + 2) \times x}{x(x - 4)(x - 4)}$$ 6. **Cancel common factors:** Cancel $x$ in numerator and denominator: $$\frac{2(x - 2)(x + 2) \times \cancel{x}}{\cancel{x}(x - 4)(x - 4)} = \frac{2(x - 2)(x + 2)}{(x - 4)^2}$$ 7. **Final simplified expression:** $$\boxed{\frac{2(x - 2)(x + 2)}{(x - 4)^2}}$$ This is the simplified form of the original expression.