All Knowledge Is Found In Blogs
There’s a page at “S.O.S. Math” on polynomial long division that looks like a pretty good introduction (in particular, there are exercises … a vital part of the process!). And here’s a post and thread on “synthetic” division in JD2718 (one of my favorite mathblogs).
For me, the main point to be made is that, until somebody understands “long” division, they’ve got no business learning how to do “synthetic” division. For this version of this class, students are free to use the “synthetic” algorithm (when appropriate) if they want to, but I’m not going to require it.