  We consider travelling wave solutions of the reaction diffusion equation with
quintic nonlinearities $u_t = u_{xx} + \mu u (1 -u )
  ( 1 +\alpha u + \beta u^2 +\gamma u^3)$. If the parameters $\alpha , \beta$
and $\gamma$ obey a special relation, then the criterion for the existence of a
strong heteroclinic connection can be expressed in terms of two of these
parameters. If an additional restriction is imposed, explicit front solutions
can be obtained. The approach used can be extended to polynomials whose highest
degree is odd.
