This equation has no real or complex solutions. One way to show that is by writing the LHS using partial fractions: 4/(x^2 -4) = 1/(x-2) - 1/(x+2). Since the term 1/(x-2) appears on both sides of the equation, it may be canceled (it’s a removable singularity), leaving the equation -1/(x+2) = 0, which only satisfied in the limit that magnitude(x) goes to infinity. Depending on whether the domain of solutions includes the point at infinity or not, there’s either no solutions (infinity excluded) or one solution in the limit that one approaches infinity.

That’s why any equation must be verified after variable are found

Excuse me , you said there are no solution, you should say : "empty set"

What conditions should we consider to claim that the equation has no solutions? I didn't get it in this case.

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-6 is one solution.