To solve the problem, we first need to simplify the expression given for \( A \):

\[
A = \frac{3x + \sqrt{3x} - 3}{x + \sqrt{2}} - \frac{\sqrt{x} - 2}{\sqrt{x} - 1} + \frac{1}{\sqrt{x} + 2} - 1
\]

### a) Rút gọn A

1. **Simplify each term**:
- Start by simplifying the first fraction:

\[
\frac{3x + \sqrt{3x} - 3}{x + \sqrt{2}}
\]

Here, combine like terms if applicable.

2. **Handle each fractional part**:
- For the term \(-\frac{\sqrt{x} - 2}{\sqrt{x} - 1}\), simplify directly.
- The last term \(\frac{1}{\sqrt{x} + 2}\) can also be left as is for now.

3. **Combine all terms**:
- Look for common denominators and add or subtract the fractions accordingly.

4. **Final simplification** should yield a single expression for \( A \).

### b) Tìm \( x \in \mathbb{Z} \) để \( A \) có giá trị nguyên

1. **Set \( A \) equal to an integer \( k \)** (where \( k \in \mathbb{Z} \)):
- Solve the equation \( A = k \).

2. **Find values of \( x \)**:
- Substitute integer values of \( x \) into the expression for \( A \) to find out when \( A \) returns an integer.
- Check \( x = 0, 1, 2, \ldots \), considering the domain of the expression to avoid undefined terms (like division by zero or negatives under square roots).

3. **Observe conditions**:
- Make sure that the values you choose for \( x \) lead to defined values for \( A \).

You can iterate through integer values for \( x \) and compute \( A \) to find eligible integers.

If you need specific calculations or further instructions, please let me know!