Let's tackle the given problem step by step.

Given:
\[ P = \frac{x}{x - 4} + \frac{1}{\sqrt{x} - 2} + \frac{1}{\sqrt{x} + 2} \]

### Part (a): Simplify \( P \)

First, let's simplify each term individually.

1. \(\frac{x}{x - 4}\) is already in its simplest form.

2. For \(\frac{1}{\sqrt{x} - 2}\) and \(\frac{1}{\sqrt{x} + 2}\), we can combine these two terms by finding a common denominator:
\[
\frac{1}{\sqrt{x} - 2} + \frac{1}{\sqrt{x} + 2} = \frac{(\sqrt{x} + 2) + (\sqrt{x} - 2)}{(\sqrt{x} - 2)(\sqrt{x} + 2)}
\]
Simplify the numerator:
\[
(\sqrt{x} + 2) + (\sqrt{x} - 2) = 2\sqrt{x}
\]
Simplify the denominator using the difference of squares:
\[
(\sqrt{x} - 2)(\sqrt{x} + 2) = x - 4
\]
So, we have:
\[
\frac{1}{\sqrt{x} - 2} + \frac{1}{\sqrt{x} + 2} = \frac{2\sqrt{x}}{x - 4}
\]

Now, combine all terms:
\[
P = \frac{x}{x - 4} + \frac{2\sqrt{x}}{x - 4}
\]
Since the denominators are the same, we can combine the numerators:
\[
P = \frac{x + 2\sqrt{x}}{x - 4}
\]

### Part (b): Calculate \( P \) when \( x = 2 - 2\sqrt{2} \)

First, let's find \(\sqrt{x}\) when \( x = 2 - 2\sqrt{2} \).

Let \( \sqrt{x} = \sqrt{2} - 1 \). Then:
\[
x = (\sqrt{2} - 1)^2 = 2 - 2\sqrt{2} + 1 = 3 - 2\sqrt{2}
\]
However, this does not match \( x = 2 - 2\sqrt{2} \). Let's try another approach.

Let \( \sqrt{x} = a - b\sqrt{2} \). Then:
\[
x = (a - b\sqrt{2})^2 = a^2 + 2b^2 - 2ab\sqrt{2}
\]
For \( x = 2 - 2\sqrt{2} \), we need:
\[
a^2 + 2b^2 = 2 \quad \text{and} \quad -2ab = -2
\]
From \(-2ab = -2\), we get \( ab = 1 \).

Substitute \( b = \frac{1}{a} \) into \( a^2 + 2b^2 = 2 \):
\[
a^2 + 2\left(\frac{1}{a}\right)^2 = 2
\]
\[
a^2 + \frac{2}{a^2} = 2
\]
Multiply by \( a^2 \):
\[
a^4 + 2 = 2a^2
\]
\[
a^4 - 2a^2 + 2 = 0
\]
Let \( y = a^2 \):
\[
y^2 - 2y + 2 = 0
\]
The discriminant is:
\[
\Delta = (-2)^2 - 4 \cdot 1 \cdot 2 = 4 - 8 = -4
\]
Since the discriminant is negative, there are no real solutions for \( a \) and \( b \).

Thus, we need to re-evaluate our approach. Let's directly substitute \( x = 2 - 2\sqrt{2} \) into the simplified form of \( P \):

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

This approach is complex due to the nested radicals. Instead, let's verify if the original problem has a simpler form or if there's a mistake in the problem statement. Given the complexity, it might be better to re-evaluate the problem or use numerical methods for specific values.

For now, the simplified form of \( P \) is:
\[
P = \frac{x + 2\sqrt{x}}{x - 4}
\]

And for \( x = 2 - 2\sqrt{2} \), further simplification might require numerical methods or a different approach to handle nested radicals.