Backslash does not use an iterative scheme, if that is what you are asking. As long as your model is linear in the parameters, a linear algebraic solution using pivoted QR is direct. No iterations at all. Some tools like lsqr are iterative. But not backslash.
You are thinking about linear least squares as if it is nonlinear least squares, where an iterative scheme is needed. Again, a model can have higher order terms in it and still be a linear least squares problem, as long as the model is linear in the coefficients.
Are you looking for a proof that it is possible to solve a (linear) least squares estimation using simple linear algebra, i.e., a QR? (Not hard to do, but not on topic for Answers.)
