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Matt_Simpson comments on Even if you have a nail, not all hammers are the same - Less Wrong
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Yes. Quadratic regression is better, often. The problem is that the number of coefficients to adjust in the model gets squared, which goes against Ockhams razor. This is precisely the problem I am working on these days, though in the context of the oil industry.
It's not too difficult to check to see if adding the extra terms improves the regression. In my original comment, I listed AIC and BIC among others. On the other hand, different diagnostics will give different answers, so there's the question of which diagnostic to trust if they disagree. I haven't learned much about regression diagnostics yet, but at the moment they all seem ad hoc (maybe because I haven't seen the theory behind them yet).