I will use trigonometric form of complex numbers.


Let  x + iy = r*(cos(a) + i*sin(a)),

where r > 0 is the modulus and "a" is a polar angle.


Then x - iy = r*(cos(a) - i*sin(a)).


According to de Moivre's theorem,

     = ,

     = .


Then

     +  =  +  = .


The problem asks me to find ONLY ONE ROOT  to equation  

    (x+iy)^5 + (x-iy)^5 = -8.     (1)


It is equivalent to find one root of the equation

     = -8.               (2)


I will take  a =  radians =  degrees = 36 degrees.


Then  cos(5a) =  = -1,  and the equation (2)  takes the form


     = -8,      

or, equivalently,

     = 8,            (3)

     = 4,             (4)

    r = .            (5)


Thus  

    x + iy =  = 


is one possible root of the equation (1).