Both answers show how to systematically check all possible factors. Teacher A makes and tests a conjecture regarding the lack of new, undiscovered factors after 8. Although Teacher A's example is successful because it shows systematic testing of all possible values of factors, it would be a tedious method to follow for larger numbers. Furthermore, it is not always possible to prove something by testing all possible values, nor is this an adequate method of proof.
Teacher B has a more sophisticated way of checking and explaining, and shows emerging understanding of factoring expressions, such as 2 x 12, to think "2 x 2 x 6" and then to write "4 x 6." Her method draws on connections to the rectangle representations to use reasoning about rearranging the area to give new factors. Her method also starts a general observation about the relationship between factors but relies on a specific case, that of an 8-by-3 rectangle.