Either quadratic or interaction term is significant in isolation, but neither are together As part of an assignment, I had to fit a model with two predictor variables. I then had to draw a plot of the models' residuals against one of the included predictors and make changes based on that. The plot showed a curvilinear trend and so I included a quadratic term for that predictor. The new model showed the quadratic term to be significant. All good so far.
However, the data suggest that an interaction makes sense, too. Adding an interaction term to the original model also 'fixed' the curvilinear trend and was also significant when added to the model (without the quadratic term). The problem is, when both the quadratic and the interaction term are added to the model, one of them is not significant.
Which term (the quadratic or the interaction) should I include in the model and why? 
 A: What makes the most sense based on the source of the data?  
We cannot answer this question for you, the computer cannot answer this question for you.  The reason that we still need statisticians instead of just statistical programs is because of questions like this.  Statistics is about more than just crunching the numbers, it is about understanding the question and the source of the data and being able to make decisions based on the science and background and other information outside the data that the computer looks at.  Your teacher is probably hoping that you will contemplate this as part of the assignment.  If I had assigned a problem like this (and I have before) I would be more interested in the justification of your answer than which you actually chose.
It is probably beyond your current class, but one approach if there is not a clear scientific reason for prefering one model over the other is model averaging, you fit both models (and maybe several other models as well), then you average together the predictions (often weighted by the goodness of fit of the different models).
Another option, when possible, is to collect more data and if possible choosing the x values so that it becomes more clear what the non-linear vs. interaction effects are.
There are some tools for comparing the fit of non-nested models (AIC, BIC, etc.), but for this case they probably will not show enough difference to overrule understanding of where the data comes from and what makes the most sense.
A: Yet another possibility, in addition to @Greg's is to include both terms, even though one is not significant. Including only statistically significant terms is not a law of the universe. 
