I would say yes.
Typically synthetic division is taught to make it seem like the algorithm only works for rational functions of the form \(\dfrac{a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0}{x-r}\), i.e. only when the divisor \(x-r\) is a monic linear polynomial.
However there are expanded algorithms that would allow for division of something like \(\dfrac{a_nx^n+\cdots+a_0}{x^2+bx+c}\) (dividing by higher order monic polynomials) as well as \(\dfrac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots+b_0}\) (dividing by any higher order polynomial provided that \(m\le n\)).
Consult the Wiki page on "synthetic division" for more info.
