1. **State the problem:** Simplify the expression $$\frac{x + 5}{x - 4} - \frac{x - 2}{x}$$ by subtracting the two rational expressions. 2. **Find a common denominator:** The denominators are $x - 4$ and $x$. The least common denominator (LCD) is $$x(x - 4)$$. 3. **Rewrite each fraction with the LCD:** $$\frac{x + 5}{x - 4} = \frac{(x + 5) \cdot x}{x(x - 4)} = \frac{x(x + 5)}{x(x - 4)}$$ $$\frac{x - 2}{x} = \frac{(x - 2)(x - 4)}{x(x - 4)}$$ 4. **Subtract the numerators over the common denominator:** $$\frac{x(x + 5) - (x - 2)(x - 4)}{x(x - 4)}$$ 5. **Expand the numerators:** $$x(x + 5) = x^2 + 5x$$ $$(x - 2)(x - 4) = x^2 - 4x - 2x + 8 = x^2 - 6x + 8$$ 6. **Subtract the expanded expressions:** $$x^2 + 5x - (x^2 - 6x + 8) = x^2 + 5x - x^2 + 6x - 8 = 11x - 8$$ 7. **Write the simplified expression:** $$\frac{11x - 8}{x(x - 4)}$$ 8. **Check for further simplification:** The numerator $11x - 8$ does not factor nicely to cancel with the denominator, so this is the simplest form. **Final answer:** $$\frac{11x - 8}{x(x - 4)}$$