I want to include the term $x$ and its square $x^2$ (predictor variables) into a regression because I assume that low values of $x$ have a positive effect on the dependent variable and high values have a negative effect. The $x^2$ should capture the effect of the higher values. I therefore expect that the coefficient of $x$ will be positive and the coefficient of $x^2$ will be negative. Besides $x$, I also include other predictor variables.
I read in some posts here that it is a good idea to center the variables in this case to avoid multicollinearity. When conducting multiple regression, when should you center your predictor variables & when should you standardize them?
Should I center both variables seperately (at the mean) or should I only center $x$ and then take the square or should I only center $x^2$ and include the original $x$?
Is it a problem if $x$ is a count variable?
In order to avoid $x$ being a count variable, I thought about dividing it by a theoretically defined area, for example 5 square kilometers. This should be a little bit similar to a point density calculation.
However, I am afraid that in this situation my initial assumption about the sign of the coefficients would not hold anymore, as when $x=2$ and $x²=4$
$x= 2 / 5 \text{ km}^2$ = $0.4 \text{ km}^2$
but $x^2$ would then be smaller because $x^2= (2/5)^2= 0.16$.