k5=5*r,

k4=-10*r^2+5*b*g*m+5*c*d*m,

k3=10*r^3-15*b*g*m*r-15*c*d*m*r+5*c*g^2*m^2+5*d^2*g*m^2+5*b^2*d*m+5*b*c^2*m,

k2=-5*r^4+15*b*g*m*r^2+15*c*d*m*r^2-10*c*g^2*m^2*r-10*d^2*g*m^2*r-10*b^2*d*m*r
   -10*b*c^2*m*r+5*d*g^3*m^3-5*b^2*g^2*m^2+5*b*c*d*g*m^2+5*c^3*g*m^2+5*b*d^3*m^2
   -5*c^2*d^2*m^2+5*b^3*c*m,

k1=r^5-5*b*g*m*r^3-5*c*d*m*r^3+5*c*g^2*m^2*r^2+5*d^2*g*m^2*r^2+5*b^2*d*m*r^2+5*b*c^2*m*r^2
   -5*d*g^3*m^3*r+5*b^2*g^2*m^2*r-5*b*c*d*g*m^2*r-5*c^3*g*m^2*r-5*b*d^3*m^2*r+5*c^2*d^2*m^2*r
   -5*b^3*c*m*r+g^5*m^4-5*b*c*g^3*m^3+5*b*d^2*g^2*m^3+5*c^2*d*g^2*m^3-5*c*d^3*g*m^3+d^5*m^3
   -5*b^3*d*g*m^2+5*b^2*c^2*g*m^2+5*b^2*c*d^2*m^2-5*b*c^3*d*m^2+c^5*m^2+b^5*m,

for the polynomial x^5-k5*x^4-k4*x^3-k3*x^2-k2*x-k1,

roots:

x1=r+b*m^(1/5)+c*m^(2/5)+d*m^(3/5)+g*m^(4/5),

x2=r+%e^((2*%i*%pi)/5)*g*m^(4/5)+%e^((4*%i*%pi)/5)*d*m^(3/5)+%e^(-(4*%i*%pi)/5)*c*m^(2/5)+%e^(-(2*%i*%pi)/5)*b*m^(1/5),

x3=r+%e^((4*%i*%pi)/5)*g*m^(4/5)+%e^(-(2*%i*%pi)/5)*d*m^(3/5)+%e^((2*%i*%pi)/5)*c*m^(2/5)+%e^(-(4*%i*%pi)/5)*b*m^(1/5),

x4=r+%e^(-(4*%i*%pi)/5)*g*m^(4/5)+%e^((2*%i*%pi)/5)*d*m^(3/5)+%e^(-(2*%i*%pi)/5)*c*m^(2/5)+%e^((4*%i*%pi)/5)*b*m^(1/5),

x5=r+%e^(-(2*%i*%pi)/5)*g*m^(4/5)+%e^(-(4*%i*%pi)/5)*d*m^(3/5)+%e^((4*%i*%pi)/5)*c*m^(2/5)+%e^((2*%i*%pi)/5)*b*m^(1/5).