The only way to solve this (or any other polynomial equation of degree 5 or greater) is by numerical approximation. I graphed it and saw that there is a root very near -2, possible multiple roots at -1, and a root at 1.
A little polynomial long division gets you to the following result:
Plugging the coefficients of the quartic factor into an on-line Quartic Solver, I got the following results:
0.5 + 1.323 i
0.5 - 1.323 i
0.5 + 1.323 i
0.5 - 1.323 i
One pair of complex conjugates with a multiplicity of 2.
John
Egw to Beta kai to Sigma
My calculator said it, I believe it, that settles it
