Positively hilarious

Let bb and cc be positive integers such that the polynomial equations x2+bx+c=0 and x2+bxc=0 x^2 + bx + c =0 \text{ and } x^2 + bx - c=0 both have integer solutions. Determine the sum of all values of b50b \leq 50 for which polynomials of this form exist.

Details and assumptions

There is no restriction on the size of cc.

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