Assuming that the dependent variable (DV) is continuous and that the other assumptions of linear regression are met, you could try any sort of robust regression (as far as I know, MM estimation seems to be the currently favored method) or quantile regression (which makes no assumptions about the residuals). With quantile regression, you'd be analyzing the median (or other quantiles) rather than the mean, but if the distribution of your DV is symmetric, that shouldn't be a problem.

Also, I'd consider looking at a model with splines rather than just linear and quadratic effects, unless there is strong theoretical reason to think that a quadratic fit is correct.