To solve the problem, let's analyze each part step by step:

### Biểu thức A:
\[ A = \frac{\sqrt{x} + 1}{x + 4\sqrt{x} + 4} \cdot \left( \frac{x}{x + 2\sqrt{x}} + \frac{x}{\sqrt{x} + 2} \right) \]

### 1) Tìm điều kiện của \( x \) để biểu thức \( A \) xác định:

To ensure \( A \) is defined, consider the following:

- \(\sqrt{x}\) must be real, so \(x \geq 0\).
- The denominator \(x + 4\sqrt{x} + 4\) must not be zero:
- Factor it: \((\sqrt{x} + 2)^2 = x + 4\sqrt{x} + 4\)
- So, \((\sqrt{x} + 2) \neq 0\), meaning \(x \neq 4\).

- \(x + 2\sqrt{x} \neq 0\) and \(\sqrt{x} + 2 \neq 0\) to avoid division by zero.

Thus, \(x \geq 0\) and \(x \neq 4\).

### 2) Rút gọn biểu thức \( A \):

We start with simplifying:
\[ \left( \frac{x}{x + 2\sqrt{x}} + \frac{x}{\sqrt{x} + 2} \right) = \frac{x(\sqrt{x} + 2) + x(x + 2\sqrt{x})}{(x + 2\sqrt{x})(\sqrt{x} + 2)} \]

Combine and simplify:
- The numerator becomes:
- \( x(\sqrt{x} + 2) + x(x + 2\sqrt{x}) = x\sqrt{x} + 2x + x^2 + 2x\sqrt{x} = x^2 + 3x\sqrt{x} + 2x \)

This matches the denominator after simplification, so the expression simplifies to:
\[ \frac{x(\sqrt{x} + 2) + x(x + 2\sqrt{x})}{(x + 2\sqrt{x})(\sqrt{x} + 2)} = 1 \]

Thus:
\[ A = \frac{\sqrt{x} + 1}{(\sqrt{x} + 2)^2} \]

### 3) Tìm \( x \) để \( |A| > A \):

This condition implies:
- \( A < 0 \)

Given:
\[ A = \frac{\sqrt{x} + 1}{(\sqrt{x} + 2)^2} < 0 \]

The expression \(\sqrt{x} + 1\) in the numerator is non-negative for \(x \geq 0\), so for \(A\) to be negative, \((\sqrt{x} + 2)^2\) must change sign, which is not possible since it’s always positive for \(x \geq 0\).

Therefore, there is no \(x\) for which \(A\) is negative. Thus, no solution satisfies \( |A| > A \) in the interval \(x \geq 0\).

### Conclusion:

Thus:
1. \(x \geq 0\) and \(x \neq 4\) for \(A\) to be defined.
2. Simplified: \( A = \frac{\sqrt{x} + 1}{(\sqrt{x} + 2)^2} \)
3. No \(x\) satisfies \( |A| > A \).