Stuck on Polynomial Division Problem Trying to find the quotient and remainder of the problem below..but keep getting to wrong. Could someone demonstrate how to solve this correctly?

It seems that you would have saved yourself a little confusion if you had simply set up the polynomial long division as follows:
$$x - 2\; \overline{|\;\;x^5 - x^4 - 4x^3 + \color{maroon}{0x^2} + 2x + 3\;}$$
and proceeded from there. (I.e., by writing in and accounting for the "implicit term" $\color{maroon}{0\cdot x^2}$, you would have kept the division properly aligned, and saved yourself a bit of a headache.)
