Finding when a equation has a solution

Hello,

  This equation in x:

    x^5 * h + x^4 * f + x^3 * f + x^2 * f + x * f + f - g = 0
  
  has a closed form solution when g=h.

  Is it possible to use maxima to find out other relationships among
  f, g and h which result in that equation having a closed form
  solution?  If so, how?

  Thanks,

  --Suresh