If `x^(4) + 2kx^(3) + x^(2) + 2kx + 1 = 0`
has exactly tow distinct positive and two distinct negative roots, then find the possible real values of k.

Explanation

To solve the equation \( x^4 + 2kx^3 + x^2 + 2kx + 1 = 0 \) and find the possible real values of \( k \) such that it has exactly two distinct positive roots and two distinct negative roots, we can follow these steps:

### Step 1: Rewrite the Equation
The given equation can be rewritten as:
\[ x^4 + 2kx^3 + x^2 + 2kx + 1 = 0 \]
We can group the terms to facilitate analysis:
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