How to solve this mathematical puzz

(x^2)+x+1+x^-1+x^-2=0

-(i)^(1/5),
(i)^(2/5),
-(i)^(3/5),
(i)^(4/5)

Express equation with common denominator of x^-2 i.e multiply through by x^2

x^4 + x^3 + x^2 + x + 1 = 0

(x^2)+x+1+x^-1+x^-2=0
or (x + 1/x)^2 + (x + 1/x) -- 2 = 0
OR (x + 1/x + 2)(x + 1/x -- 1) = 0
Giving x^2 + 2x + 1 = 0 and x^2 -- x + 1 = 0
whence x = -- 1, -- 1, 1/2 +/-- i sqrt(3)/2 ANSWER