To solve the problem, we'll simplify each component of the expression step by step.

### a) Rút gọn biểu thức \(P\).

The original expression is:

\[
P = \left(\frac{3\sqrt{x}}{\sqrt{x} - 1} - \frac{2\sqrt{x}}{3x + 2\sqrt{x}} - \frac{9x + \sqrt{x} + 1}{3x - \sqrt{x} - 2}\right) \cdot \frac{3\sqrt{x} + 1}{7x - 7\sqrt{x}}
\]

#### Simplification

1. **LCD of the fractions**:
- The expressions \(\frac{3\sqrt{x}}{\sqrt{x} - 1}\), \(\frac{2\sqrt{x}}{3x + 2\sqrt{x}}\), and \(\frac{9x + \sqrt{x} + 1}{3x - \sqrt{x} - 2}\) all need a common denominator. Let's find each term:

2. **Simplify each fraction**:
- Each term should be worked out by obtaining a least common denominator (LCD), usually the product of the individual denominators, and adjusting numerators accordingly.

3. **Combining the terms inside the parentheses**:
- Combine the simplified expressions inside the large parentheses.

4. **Simplifying the second fraction**:

- The fraction \(\frac{3\sqrt{x} + 1}{7x - 7\sqrt{x}}\) can be simplified by factorization:
\[
= \frac{3\sqrt{x} + 1}{7(\sqrt{x})(\sqrt{x} - 1)}
\]

5. **Final simplification**:
- Substitute back the simplified expressions, then perform the multiplication of fractions:
\[
P = (\text{Simplified Inner Expression}) \cdot \frac{3\sqrt{x} + 1}{7(\sqrt{x})(\sqrt{x} - 1)}
\]
- Simplify numerators with common factors as much as possible.

### b) Tìm \(x\) sao cho \(P\) nhận giá trị là một số nguyên.

After reducing, inspect which values of \(x\) will result in \(P\) being an integer. This might lead to solving an equation involving integer solutions or analyzing the behavior of P when x equals non-zero integers that satisfy the expression as a whole.

1. **Test Integer Solutions**:
- Test possible integer values especially for \(x > 0\) that can result in the coefficients cancelling out resulting in integer \(P\).

2. **Analyzing integer output**:
- Rearrange as necessary and solve any equations that guarantee integer results from simplified \(P\).

Evaluating specific examples and plugging back results can concretely establish integer values and solutions. Comprehensive checks per transformed expression may further assist final answer selection and verification.