To simplify the given expression, we can follow these steps:

1. **Separate the numerator and the denominator** of the main expression.
2. **Simplify each part** of the expression.
3. **Combine and simplify** the final result.

Let's break it down step-by-step.

### Step 1: Identify the Expression
The expression is:

\[
\frac{\left( \frac{\sqrt{x} + 4}{x - 2\sqrt{x}} + \frac{3}{\sqrt{x} - 2} \right)}{\left( \frac{\sqrt{x}}{\sqrt{x} - 2} - \frac{\sqrt{x} + 2}{\sqrt{x}} \right)}
\]

### Step 2: Simplify the Numerator

**Numerator:**

1. First part: \( \frac{\sqrt{x} + 4}{x - 2\sqrt{x}} \)

Notice that \( x - 2\sqrt{x} = (\sqrt{x})^2 - 2\sqrt{x} = (\sqrt{x} - 2)^2 \).

So, the first part simplifies to:

\[
\frac{\sqrt{x} + 4}{(\sqrt{x} - 2)^2}
\]

2. Second part: \( \frac{3}{\sqrt{x} - 2} \)

Now we combine both fractions:

- To combine \( \frac{\sqrt{x} + 4}{(\sqrt{x} - 2)^2} \) and \( \frac{3}{\sqrt{x} - 2} \), we need a common denominator, which is \( (\sqrt{x} - 2)^2 \):

\[
\frac{\sqrt{x} + 4 + 3(\sqrt{x} - 2)}{(\sqrt{x} - 2)^2}
\]

Simplifying the numerator:
\[
\sqrt{x} + 4 + 3\sqrt{x} - 6 = 4\sqrt{x} - 2
\]

So the entire numerator simplifies to:

\[
\frac{4\sqrt{x} - 2}{(\sqrt{x} - 2)^2}
\]

### Step 3: Simplify the Denominator

**Denominator:**

1. First part: \( \frac{\sqrt{x}}{\sqrt{x} - 2} \)
2. Second part: \( \frac{\sqrt{x} + 2}{\sqrt{x}} \)

To combine these, we find a common denominator, which is \( \sqrt{x}(\sqrt{x} - 2) \). Thus, we have:

\[
\frac{\sqrt{x}^2}{\sqrt{x}(\sqrt{x} - 2)} - \frac{(\sqrt{x} + 2)(\sqrt{x} - 2)}{\sqrt{x}(\sqrt{x} - 2)}
\]

The second part expands and simplifies as:

\[
(\sqrt{x})^2 - 4 = x - 4
\]

Combining gives us:

\[
\frac{x - (x - 4)}{\sqrt{x}(\sqrt{x} - 2)} = \frac{4}{\sqrt{x}(\sqrt{x} - 2)}
\]

### Step 4: Final Expression

So now we have:

\[
\frac{\frac{4\sqrt{x} - 2}{(\sqrt{x} - 2)^2}}{\frac{4}{\sqrt{x}(\sqrt{x} - 2)}}
\]

Now, multiply by the reciprocal:

\[
\frac{(4\sqrt{x} - 2) \cdot \sqrt{x}(\sqrt{x} - 2)}{(\sqrt{x} - 2)^2 \cdot 4}
\]

Cancelling \(4\) in the numerator and denominator gives us:

\[
\frac{(4\sqrt{x} - 2) \cdot \sqrt{x}(\sqrt{x} - 2)}{(\sqrt{x} - 2)^2}
\]

### Step 5: Final Simplification

We can simplify the fraction by observing:

\[
\frac{(4\sqrt{x} - 2)\sqrt{x}}{\sqrt{x} - 2}
\]

One more step to check simplifications could yield:

When \( \sqrt{x} - 2 \) cancels further, we get:

### Conclusion:

Thus, the simplified expression is

\[
\frac{(4\sqrt{x} - 2)\sqrt{x}}{(\sqrt{x} - 2)}
\]

To simplify further:

\[
= \frac{2\sqrt{x}(2\sqrt{x} - 1)}{(\sqrt{x} - 2)}
\]

Final answer may depend on constraints for \(x - 2\) not zero etc. Simplifying also returns consistent results with \(x\) in range of 0 to constraints.

This summarizes the steps taken. Would you like to verify specific numerical values for \(x\) so we can further confirm or determine any constraints or specific evaluations?