Why is it that the most general form of the remainder in a polynomial is given by R(x) = ax +b?
If the remainder were a polynomial of degree greater $\displaystyle n$ than $\displaystyle 1$ then it can be rewriten in the form:
$\displaystyle R(x)=(x+1)(x-4)P_{n-2} (x) +Q_{n-1}(x)$
Where $\displaystyle P_{n-2}$ and $\displaystyle Q_{n-1}$ are polynomials of degree $\displaystyle n-2$ and no more than $\displaystyle n-1$ respectivly, so it would not have been the remainder.