I've always found teaching common factoring to be a little troublesome, especially having them get the correct quotient in the brackets.. I stumbled on a technique that seemed to work very well for one of my students.

We started by factoring a a few numbers, and then went on to finding GCF. We then spent some time simplifying rational expressions, where all the exponents would be positive or 0, then we put them together to find the GCF of two monomials. This much is pretty obvious scaffolding. The trick that seemed to make getting the final expression right was to write it as a rational expression and simplify. For example, to common factor 12x^3+8x^2, the student would find the GCF of 4x^2, the write

12x^3 + 8x^2

---------------

4x^2

= 3x + 2

and as a final step put the divisor in brackets and write the quotient in front: 4x^2(3x + 2). Doing it this way really seemed to click with him. I'm looking forward to trying it with other students.