I have an example from a paper (listed below) that I cannot figure out. I can divide normal polynomials, but the alternative ways to divide Laurent polynomials is beyond me at the moment. The paper lists two polynomials, and three possible quotient/remainder pairs. The polynomials are,
\begin{equation} a(z) = z^{-1} + 6 + z, \quad b(z) = 4 + 4z, \end{equation}
and the three sets of quotients and remainders are,
\begin{equation} q(z) = \frac{1}{4}(z^{-1} + 5) , \quad r(z) = -4z, \end{equation}
\begin{equation} q(z) = \frac{1}{4}(z^{-1} + 1), \quad r(z) = 4, \end{equation}
\begin{equation} q(z) = \frac{1}{4}(5z^{-1} + 1), \quad r(z) = -4z^{-1}. \end{equation}
I understand how to obtain the last quotient and remainder, but I do not understand how the first two were obtained. A worked example of one or the other would be fantastic, thanks. This is from the Daubechies and Sweldens paper on factoring wavelet transforms.