Say you are trying to factor this . . .
You start by putting 2x in the front of each binomial. This feels wrong. It should.
So, to cancel out the extra 2, you divide by 2 right at the beginning. This was my teensy tweak. I used to have students do this at the end. It makes more sense here, and students are less likely to forget.
Then continue normally, multiply a and c. Find two numbers that multiply to get ac and add to get b. Put these at the back of each binomial.
Then, with the two already hanging out there, you just look for which binomial can be reduced by 2.
In this situation, the 6 has to be split into 3 and 2 before reducing.
Students didn't have any trouble with that.
Disclaimer: This is not my method. I saw it first while observing another teacher. I have seen others write about it as well, so I am not sure where to give credit.