1. **Problem statement:** Simplify the expression $$A = \left( \frac{x + \sqrt{x}}{\sqrt{x} + 1} \right) \times \left( \frac{\sqrt{x} - x}{\sqrt{x} - 1} \right) \times \left( 1 + \frac{1}{\sqrt{x}} \right)$$ for $x > 0$, $x \neq 1$. Then find the values of $x$ such that $A = 4$. 2. **Step 1: Simplify each part.** - Let $t = \sqrt{x}$, so $x = t^2$ and $t > 0$, $t \neq 1$. - Rewrite $A$ in terms of $t$: $$A = \left( \frac{t^2 + t}{t + 1} \right) \times \left( \frac{t - t^2}{t - 1} \right) \times \left( 1 + \frac{1}{t} \right)$$ 3. **Step 2: Simplify each fraction:** - First fraction: $$\frac{t^2 + t}{t + 1} = \frac{t(t + 1)}{t + 1} = t \quad \text{(since } t + 1 \neq 0 \text{)}$$ - Second fraction: $$\frac{t - t^2}{t - 1} = \frac{t(1 - t)}{t - 1} = \frac{t(1 - t)}{t - 1}$$ Note that $1 - t = -(t - 1)$, so: $$\frac{t(1 - t)}{t - 1} = \frac{t \times (-(t - 1))}{t - 1} = -t$$ - Third term: $$1 + \frac{1}{t} = \frac{t + 1}{t}$$ 4. **Step 3: Multiply all simplified parts:** $$A = t \times (-t) \times \frac{t + 1}{t} = (-t^2) \times \frac{t + 1}{t} = -t^2 \times \frac{t + 1}{t}$$ Simplify: $$-t^2 \times \frac{t + 1}{t} = -t (t + 1) = -t^2 - t$$ 5. **Step 4: Express $A$ back in terms of $x$:** $$A = -x - \sqrt{x}$$ 6. **Step 5: Find $x$ such that $A = 4$:** $$-x - \sqrt{x} = 4$$ Multiply both sides by $-1$: $$x + \sqrt{x} = -4$$ Since $x > 0$ and $\sqrt{x} > 0$, the left side is positive, but the right side is negative. This is impossible. **Therefore, there is no $x > 0$, $x \neq 1$ such that $A = 4$.**