hey i have a good solution for this
1+x+x2+x3+x4=0
(x^5-1)/(x-1)=0
here the roots are p,p^2,p^3,p^4
where p is the fifth root of unity
p^(5+n)=p^n
like we do in cube root of unity
using this property the roots will again become p^2,p^4,p,p^3
so the equation will become the same & a+b+c+d=4