1. **State the problem:** Simplify the expression $$x = -\frac{1}{2}(a+b)\left(\frac{1}{ab}+1\right) \pm \left(\frac{1}{ab}+1\right)\sqrt{(a+b)^2 - 4ab}$$. 2. **Rewrite the terms inside the parentheses:** $$\frac{1}{ab} + 1 = \frac{1}{ab} + \frac{ab}{ab} = \frac{1+ab}{ab}$$ 3. **Substitute back:** $$x = -\frac{1}{2}(a+b) \cdot \frac{1+ab}{ab} \pm \frac{1+ab}{ab} \sqrt{(a+b)^2 - 4ab}$$ 4. **Simplify the square root:** $$(a+b)^2 - 4ab = a^2 + 2ab + b^2 - 4ab = a^2 - 2ab + b^2 = (a - b)^2$$ 5. **Replace the square root:** $$\sqrt{(a+b)^2 - 4ab} = \sqrt{(a-b)^2} = |a-b|$$ 6. **Rewrite the expression:** $$x = -\frac{1}{2}(a+b) \cdot \frac{1+ab}{ab} \pm \frac{1+ab}{ab} |a-b|$$ 7. **Factor out common term:** $$x = \frac{1+ab}{ab} \left(-\frac{1}{2}(a+b) \pm |a-b| \right)$$ 8. **Final simplified form:** $$x = \frac{1+ab}{ab} \left(-\frac{a+b}{2} \pm |a-b| \right)$$ This is the simplified expression for $x$ in terms of $a$ and $b$.