Let \(b\) and \(c\) be **positive** integers such that the polynomial equations \[ x^2 + bx + c =0 \text{ and } x^2 + bx - c=0\] both have integer solutions. Determine the sum of all values of \(b \leq 50\) for which polynomials of this form exist.

**Details and assumptions**

There is no restriction on the size of \(c\).

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