I will assume that you are talking about factoring quadratics, am I right?
A lot of teachers use the 'guess and check' method especially when the leading coefficient is 1. This takes a little more time when that coefficient is something other than 1.
Here's how I approach those. You might find it useful.
First a simple one:
Step 1: Multiply the leading coefficient (2) times the constant (3) to get (6).
Step 2: Determine (mentally) what two factors make (6) that also add up to the middle coefficient (7). You can quickly come up with (6) and (1).
Step 3: Replace the original middle term with these two values from Step 2.
Step 4: Factor by grouping
Here's a second one:
Step 1: 6 times -6 = -36
Step 2: With a little thinking, I can come up with -4 and 9 as my two factors that add up to +5.
I hope this is helpful to you and your students. It works for mine.