The sum of all real solutions of the equation \[\sqrt{(x+1)(x-1)^3}-2x\sqrt{x-2} =0\] can be expressed as \(\sqrt{a}+\sqrt{b}+\sqrt{c}\), where \(a,b,\) and \(c\) are positive integers. What is the value of \(a+b+c\)?

This problem is posed by Aram T.

**Details and assumptions**

For this question, the domain of the square root function is non-negative numbers.

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