To solve the expressions in your query, let's clarify the steps needed for each one while determining the appropriate conditions for each variable involved.

### Expression 3:
\[
A = \frac{x + 3\sqrt{x}}{\sqrt{x} + 3} \cdot \left( \frac{x - 1}{\sqrt{x - 1} + 1} \right)
\]

**Conditions:**
1. \(x \geq 0\) (since \(\sqrt{x}\) is involved).
2. \(x \neq 1\) (to avoid division by zero).

**Rationalizing:**
For simplification, factor or multiply terms where necessary, then simplify by cancelling common factors.

### Expression 5:
\[
A = \frac{\sqrt{a - 2}}{\sqrt{a + 2}} \cdot \sqrt{a - 4}
\]

**Conditions:**
1. \(a \geq 4\) (to ensure all square roots are defined).

**Rationalizing:**
Multiply out the terms and simplify by noting that \(\sqrt{a - 2}\) and \(\sqrt{a + 2}\) can be expressed in terms of one radical if they share a common base.

### Expression 4:
\[
A = \left( \frac{\sqrt{a - 1}}{\sqrt{a + 1}} - \frac{1}{\sqrt{a - 1}} \right) \cdot \left( -\frac{2}{a + 1} \right)^2
\]

**Conditions:**
1. \(a \geq 1\) (to ensure all square roots are defined).

**Rationalizing:**
Calculate the fraction and simplify the expressions by combining like terms.

### Expression 6:
\[
A = \frac{5\sqrt{x} - 3}{\sqrt{x - 2}} + \frac{3}{\sqrt{x + 2}} + \frac{x^2 + 2\sqrt{x} + 8}{x - 4}
\]

**Conditions:**
1. \(x \geq 2\) (to ensure all square roots are defined).
2. \(x \neq 4\) (to avoid division by zero).

**Rationalizing:**
Find a common denominator and combine the values accordingly.

### Summary
For all expressions, ensure the defined conditions are considered for variable \(x\) or \(a\) to maintain the validity of the square roots and prevent any division by zero. After applying these conditions, the next steps involve rationalizing and simplifying the expressions respectively. Let me know if you need step-by-step solutions for any specific expression!