1. **Problem statement:** Simplify the expression $$A = \frac{1}{\sqrt{x} + 1} + \frac{x}{\sqrt{x} - x}$$. 2. **Recall important rules:** - To simplify expressions with radicals, rationalize denominators when possible. - Factor expressions to find common terms. 3. **Rewrite the second denominator:** Note that $$x = (\sqrt{x})^2$$, so $$\sqrt{x} - x = \sqrt{x} - (\sqrt{x})^2$$. 4. **Simplify the second fraction:** $$\frac{x}{\sqrt{x} - x} = \frac{(\sqrt{x})^2}{\sqrt{x} - (\sqrt{x})^2}$$. 5. **Factor denominator:** $$\sqrt{x} - (\sqrt{x})^2 = \sqrt{x}(1 - \sqrt{x})$$. 6. **Rewrite the fraction:** $$\frac{(\sqrt{x})^2}{\sqrt{x}(1 - \sqrt{x})} = \frac{\sqrt{x} \cancel{\sqrt{x}}}{\cancel{\sqrt{x}} (1 - \sqrt{x})} = \frac{\sqrt{x}}{1 - \sqrt{x}}$$. 7. **Rewrite the first fraction:** $$\frac{1}{\sqrt{x} + 1}$$. 8. **Find common denominator:** The denominators are $$\sqrt{x} + 1$$ and $$1 - \sqrt{x}$$. Note that $$1 - \sqrt{x} = -(\sqrt{x} - 1)$$ and $$\sqrt{x} + 1 = (\sqrt{x} + 1)$$. 9. **Multiply numerator and denominator of the first fraction by $$1 - \sqrt{x}$$:** $$\frac{1}{\sqrt{x} + 1} \times \frac{1 - \sqrt{x}}{1 - \sqrt{x}} = \frac{1 - \sqrt{x}}{(\sqrt{x} + 1)(1 - \sqrt{x})}$$. 10. **Multiply numerator and denominator of the second fraction by $$\sqrt{x} + 1$$:** $$\frac{\sqrt{x}}{1 - \sqrt{x}} \times \frac{\sqrt{x} + 1}{\sqrt{x} + 1} = \frac{\sqrt{x}(\sqrt{x} + 1)}{(1 - \sqrt{x})(\sqrt{x} + 1)}$$. 11. **Add the two fractions:** $$A = \frac{1 - \sqrt{x}}{(\sqrt{x} + 1)(1 - \sqrt{x})} + \frac{\sqrt{x}(\sqrt{x} + 1)}{(1 - \sqrt{x})(\sqrt{x} + 1)} = \frac{1 - \sqrt{x} + \sqrt{x}(\sqrt{x} + 1)}{(\sqrt{x} + 1)(1 - \sqrt{x})}$$. 12. **Simplify numerator:** $$1 - \sqrt{x} + \sqrt{x} \cdot \sqrt{x} + \sqrt{x} \cdot 1 = 1 - \sqrt{x} + x + \sqrt{x} = 1 + x$$. 13. **Simplify denominator:** $$(\sqrt{x} + 1)(1 - \sqrt{x}) = 1 - (\sqrt{x})^2 = 1 - x$$. 14. **Final simplified expression:** $$A = \frac{1 + x}{1 - x}$$. **Answer:** $$\boxed{\frac{1 + x}{1 - x}}$$