To simplify the expression \( P \), let's analyze each term:

Given:
\[
P = \left( x^4 + \frac{4x}{x^2 - 4} \right) \div \left( \frac{x+2}{2x-4} + \frac{2x - 3x}{x^3 - 4x} - \frac{x-4}{x-2} \right)
\]

### Step 1: Simplify Each Term
- **First term**:
\[
x^4 + \frac{4x}{x^2 - 4} = x^4 + \frac{4x}{(x-2)(x+2)}
\]

- **Second term (denominator)**:
\[
\frac{x+2}{2x-4} + \frac{2x - 3x}{x^3 - 4x} - \frac{x-4}{x-2}
\]

- Simplify the fractions:
- \(\frac{x+2}{2x-4} = \frac{x+2}{2(x-2)}\)
- \(\frac{2x-3x}{x^3 - 4x} = \frac{-x}{x(x^2 - 4)} = \frac{-1}{x^2 - 4}\)
- \(\frac{x-4}{x-2}\)

### Step 2: Find a Common Denominator for the Denominator
- The least common denominator for the second term is \((x-2)(x+2)\).

- Rewrite each fraction:
- \(\frac{x+2}{2(x-2)} = \frac{x+2}{2(x-2)} \cdot \frac{x+2}{x+2} = \frac{(x+2)^2}{2(x-2)(x+2)}\)
- \(\frac{-1}{x^2-4} = \frac{-1}{(x-2)(x+2)}\)
- \(\frac{x-4}{x-2} = \frac{x-4}{x-2} \cdot \frac{x+2}{x+2} = \frac{(x-4)(x+2)}{(x-2)(x+2)}\)

Combine the fractions:
\[
\frac{(x+2)^2 - 2 - (x-4)(x+2)}{2(x-2)(x+2)}
= \frac{x^2 + 4x + 4 - 2 - (x^2 - 4x - 8)}{2(x-2)(x+2)}
= \frac{x^2 + 4x + 4 - 2 - x^2 + 4x + 8}{2(x-2)(x+2)}
= \frac{8x + 10}{2(x-2)(x+2)}
= \frac{4x + 5}{(x-2)(x+2)}
\]

### Step 3: Simplify \( P \)
Combine the entire expression:
\[
P = \left( x^4 + \frac{4x}{(x-2)(x+2)} \right) \div \left( \frac{4x + 5}{(x-2)(x+2)} \right)
\]

Rationalizing the division:
\[
P = \left( x^4 + \frac{4x}{(x-2)(x+2)} \right) \cdot \frac{(x-2)(x+2)}{4x + 5}
\]
\[
P = \frac{x^4(x-2)(x+2) + 4x}{4x + 5}
\]

Continue simplifying:
Multiply out and simplify as needed based on the original specific expression. Make sure no steps of simplification lead to undefined expressions or division by zero.

### Final Step: Review and Simplify Any Possible Further
Without values for \(x\), absolute simplification beyond this relies on evaluating specific conditions or inputs.