1. **State the problem:** Simplify the expression $$\frac{x+1}{x-1} - \frac{1}{x} + x + \frac{1}{(x-1)^2}.$$\n\n2. **Find a common denominator:** The denominators are $x-1$, $x$, and $(x-1)^2$. The least common denominator (LCD) is $$x(x-1)^2.$$\n\n3. **Rewrite each term with the LCD:**\n- First term: $$\frac{x+1}{x-1} = \frac{(x+1) \cdot x (x-1)}{x (x-1)^2} = \frac{(x+1) x (x-1)}{x (x-1)^2}.$$\n- Second term: $$-\frac{1}{x} = -\frac{(x-1)^2}{x (x-1)^2}.$$\n- Third term: $$x = \frac{x \cdot x (x-1)^2}{x (x-1)^2} = \frac{x^2 (x-1)^2}{x (x-1)^2}.$$\n- Fourth term: $$\frac{1}{(x-1)^2} = \frac{x}{x (x-1)^2}.$$\n\n4. **Combine all terms over the common denominator:**\n$$\frac{(x+1) x (x-1) - (x-1)^2 + x^2 (x-1)^2 + x}{x (x-1)^2}.$$\n\n5. **Expand and simplify the numerator:**\n- Expand $(x+1) x (x-1)$: $$x(x+1)(x-1) = x(x^2 -1) = x^3 - x.$$\n- Expand $(x-1)^2$: $$x^2 - 2x + 1.$$\n- Expand $x^2 (x-1)^2$: $$x^2 (x^2 - 2x + 1) = x^4 - 2x^3 + x^2.$$\n\nSo numerator becomes:\n$$x^3 - x - (x^2 - 2x + 1) + x^4 - 2x^3 + x^2 + x.$$\n\n6. **Simplify numerator by combining like terms:**\n$$x^4 + (x^3 - 2x^3) + (-x^2 + x^2) + (-x + 2x + x) - 1 = x^4 - x^3 + 2x - 1.$$\n\n7. **Final simplified expression:**\n$$\frac{x^4 - x^3 + 2x - 1}{x (x-1)^2}.$$