How to find damping ratio of a 4th order system?
The damping ratio is a parameter, usually denoted by ζ (zeta),1 that
  characterizes the frequency response of a second order ordinary
  differential equation.


The quote above is taken from Wikipedia: Damping ratio. In other words it relates to a 2nd order transfer function and not a 4th order system. Having said that, if it is possible to reduce the denominator to two multiplying equations each of the form: -

\$s^2 +2s\zeta \omega_n + \omega_n^2\$ (where \$\zeta\$ is damping ratio and \$\omega_n\$ is natural resonant frequency)

Then it would have some meaning.