To simplify the expression

\[
K = \frac{(x + 1)}{(x^{2} - 4)} - \frac{x}{(x^{2} + 4x + 4)} : \frac{x}{(x^{3} + 2x^{2} - 4x - 8)}
\]

we'll follow these steps:

1. **Factor the denominators**:
- \(x^2 - 4 = (x - 2)(x + 2)\)
- \(x^2 + 4x + 4 = (x + 2)^2\)
- \(x^3 + 2x^2 - 4x - 8 = (x + 2)(x^2 - 4) = (x + 2)(x - 2)(x + 2) = (x + 2)^2(x - 2)\)

2. **Rewrite the expression**:
\[
K = \frac{x + 1}{(x - 2)(x + 2)} - \frac{x}{(x + 2)^2} \div \frac{x}{(x + 2)^2(x - 2)}
\]

3. **Rewrite the division as multiplication**:
\[
K = \frac{x + 1}{(x - 2)(x + 2)} - \frac{x}{(x + 2)^2} \cdot \frac{(x + 2)^2(x - 2)}{x}
\]
Simplifying gives:
\[
K = \frac{x + 1}{(x - 2)(x + 2)} - (x - 2)
\]

4. **Find a common denominator**:
The common denominator for the two fractions is \((x - 2)(x + 2)\). Rewrite \((x - 2)\) to have the same denominator:
\[
K = \frac{x + 1}{(x - 2)(x + 2)} - \frac{(x - 2)(x + 2)(x - 2)}{(x - 2)(x + 2)}
\]
\[
K = \frac{x + 1 - (x - 2)(x + 2)}{(x - 2)(x + 2)}
\]

5. **Expand and simplify the numerator**:
\[
(x - 2)(x + 2) = x^2 - 4
\]
Thus,
\[
K = \frac{x + 1 - (x^2 - 4)}{(x - 2)(x + 2)} = \frac{x + 1 - x^2 + 4}{(x - 2)(x + 2)} = \frac{-x^2 + x + 5}{(x - 2)(x + 2)}
\]

6. **Rearrange the numerator**:
\[
K = \frac{-(x^2 - x - 5)}{(x - 2)(x + 2)}
\]

This is the simplified form of the expression. To summarize:

\[
K = \frac{-(x^2 - x - 5)}{(x - 2)(x + 2)}
\]