Để rút gọn biểu thức \( M \), ta thực hiện các bước sau:

Biểu thức ban đầu:
\[ M = \left( \frac{\sqrt{x}}{x - 1} + \frac{\sqrt{x}}{\sqrt{x} - 1} \right) \cdot \left( \frac{2}{x} - \frac{2 - x}{x + x \sqrt{x}} \right) \]

Bước 1: Rút gọn phần đầu tiên của biểu thức:
\[ \frac{\sqrt{x}}{x - 1} + \frac{\sqrt{x}}{\sqrt{x} - 1} \]

Ta có thể quy đồng mẫu số:
\[ \frac{\sqrt{x}}{x - 1} + \frac{\sqrt{x}}{\sqrt{x} - 1} = \frac{\sqrt{x}(\sqrt{x} - 1) + \sqrt{x}(x - 1)}{(x - 1)(\sqrt{x} - 1)} \]

\[ = \frac{x - \sqrt{x} + x\sqrt{x} - \sqrt{x}}{(x - 1)(\sqrt{x} - 1)} \]

\[ = \frac{x + x\sqrt{x} - 2\sqrt{x}}{(x - 1)(\sqrt{x} - 1)} \]

Bước 2: Rút gọn phần thứ hai của biểu thức:
\[ \frac{2}{x} - \frac{2 - x}{x + x \sqrt{x}} \]

Ta có thể quy đồng mẫu số:
\[ \frac{2}{x} - \frac{2 - x}{x(1 + \sqrt{x})} = \frac{2(1 + \sqrt{x}) - (2 - x)}{x(1 + \sqrt{x})} \]

\[ = \frac{2 + 2\sqrt{x} - 2 + x}{x(1 + \sqrt{x})} \]

\[ = \frac{x + 2\sqrt{x}}{x(1 + \sqrt{x})} \]

\[ = \frac{\sqrt{x}(x + 2\sqrt{x})}{x\sqrt{x}(1 + \sqrt{x})} \]

\[ = \frac{\sqrt{x}(x + 2\sqrt{x})}{x\sqrt{x}(1 + \sqrt{x})} = \frac{x + 2\sqrt{x}}{x(1 + \sqrt{x})} \]

Bước 3: Kết hợp hai phần đã rút gọn:
\[ M = \left( \frac{x + x\sqrt{x} - 2\sqrt{x}}{(x - 1)(\sqrt{x} - 1)} \right) \cdot \left( \frac{x + 2\sqrt{x}}{x(1 + \sqrt{x})} \right) \]

\[ = \frac{(x + x\sqrt{x} - 2\sqrt{x})(x + 2\sqrt{x})}{(x - 1)(\sqrt{x} - 1)x(1 + \sqrt{x})} \]

\[ = \frac{x^2 + 2x\sqrt{x} + x^2\sqrt{x} + 2x - 2x\sqrt{x} - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)x(1 + \sqrt{x})} \]

\[ = \frac{x^2 + x^2\sqrt{x} + 2x - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)x(1 + \sqrt{x})} \]

\[ = \frac{x(x + x\sqrt{x} + 2 - 4\sqrt{x})}{(x - 1)(\sqrt{x} - 1)x(1 + \sqrt{x})} \]

\[ = \frac{x(x + x\sqrt{x} + 2 - 4\sqrt{x})}{(x - 1)(\sqrt{x} - 1)x(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4\sqrt{x}}{(x - 1)(\sqrt{x} - 1)(1 + \sqrt{x})} \]

\[ = \frac{x + x\sqrt{x} + 2 - 4