Positively hilarious

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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