So I will start off by saying that you could add as many polynomial terms and interactions as you want until you get a near perfect $R^2$, so that in an of itself doesn't make it a correct model. But how to display those complicated non-linear effects and interactions is a good question. Of course a table with much fewer of those terms are impossible to interpret - let alone one with hundreds of terms.
With three variables though our problem is not so bad if we simply plot the predictions of the model. So here I generated three sets of variables,
X1, X2 & X3 from a normal distribution with a mean of 0 and a variance of 1. I then created a variable
Y that is the following function of the X's:
COMPUTE Y = 5 + 0.6*(X1) + 0.2*(X2) + -3*(X3) +
0.38*(X1**2) + 0.15*(X2**2) + -0.1*(X3**2) +
0.3*(X1*X2) + -0.1*(X2*X3) + RV.NORMAL(0,1).
This includes quadratic terms and interactions.
RV.NORMAL(0,1) is the SPSS statement for generative a random variable normally distributed with a mean of 0 and a standard deviation of 1 - so
Y has a bit of random error. Here is what the estimated regression coefficients look like:
So that is only helpful for hypothesis testing, not really understanding the predictions our model is making. So what I did to plot the predictions was create a new set of data with all of the X's regularly distributed between -2 and 2 with intervals of .4. I then create all of the polynomial and interaction terms and then score the model on this new data, generate predictions, and then create line plots of those predictions.
To interpret this plot, the Y axis plots the predicted value of Y from the model. The X axis is the value of X1, the panels are the values of X3 in the .4 increments, and the separate lines correspond to the same increments for X2. I chose a continuous color ramp for X2, as matching the exact interval to the colors is difficult. But you can see here it is a smooth function, where green corresponds to low values of X2 and purple corresponds to high values of X2.
This concept of generating predicted values of the model over reasonable sets of covariates extends to basically any type of regression model. My experience is that many models with more complicated polynomial and interaction terms tend to produce similar predictions over typical covariate values - it is in the tails that the predictions diverge (which you should be wary of extrapolating to the tails in most circumstances anyway).