To solve this problem, let's go through each part step by step:

### a) Rút gọn \( P \)

Given:

\[
P = \left(\frac{x + 2}{x\sqrt{x} - 1} + \frac{\sqrt{x}}{x + \sqrt{x} + 1} + \frac{1}{1 - \sqrt{x}}\right) \cdot \frac{\sqrt{x} - 1}{2}
\]

1. **Simplify each fraction**: Start by finding a common denominator for the terms inside the parentheses.
2. **Multiply by \(\frac{\sqrt{x} - 1}{2}\)**: Distribute this term after simplifying the expression inside the parentheses.

### b) Tìm các giá trị của \( x \) để \( P = \frac{2}{7} \)

After simplifying \( P \), set it equal to \(\frac{2}{7}\) and solve for \( x \):

1. **Solve the equation**: Rearrange the equation to isolate terms involving \( x \).
2. **Check domain restrictions**: Ensure \( x \geq 0\) and \( x \neq 1\).

### c) So sánh \( 2P \) và \( P^2 \)

We need to compare \( 2P \) and \( P^2 \):

1. **Formula for comparison**: \( 2P \) is twice the value of \( P\), while \( P^2 \) is \( P\) squared.
2. **Solve inequality**: Determine if \( 2P > P^2 \) or \( 2P < P^2 \) by plugging in values or further simplifying.

Go through each step carefully, performing algebraic manipulations and checking for potential restrictions or special cases based on the values of \( x \).