[SOLVED] Regression of many variables

I have a pretty good understanding of regressions from high school statistics, with one independent variable and one dependent variable.

I would like to create a regression with a vector of N independent variables and a vector of M dependent variables. At this point I don't know what shape my data will be, so I figure a high-order polynomial regression will work. Maybe something of this form:

Is this something that is done very often or am I on the wrong track? I would be happy to just get some links.

Thanks,

Neal