First, you get the students accustomed to multiplying binomials using the box method, and factoring basic quadratics by using the box method backwards. I've made notes (download) that walk you through both processes.
Then you teach the students this trick, which I call "x marks the spot" for no particular reason: I give the students a series of large X marks, each with a number above it and a number below it. The students have to write numbers on either side of the X that add to make the top value, and that multiply to make the bottom value. For example:
In this case, you want two numbers that will add to make 10, and multiply to make 16.
Yes indeed, they are 8 and 2. So you would write 8 on the left, and 2 on the right, or the other way around. Students love these (here's a page of them).
Once you've got everybody comfortable with X marks the spot and with using the box, you put these two pieces together, and you get a technique that will directly factor any quadratic, or determine that it's not factorable, if that's the case.
Let's say you want to factor 6x2 – 11x – 10. First you make an X :
The number on the top of the X is the value of "b" in the quadratic expression; the number on the bottom of the X is the value of "ac" in the quadratic expression (6 * -10 = -60).
We find the other two numbers as before.
Now we bring in the box.
As always, the first and last term of the quadratic are loaded into the first and last cells in the box.
Now, the numbers we got in the X (the -15 and the 4) become the coefficients on the linear parts.
The box is full. We can now start anywhere- with any row, or any column- and extract the greatest common factor. We write this outside the box.
The rest of it falls into place.
We have it now: 6x2 – 11x – 10 = (2x - 5)(3x + 2) .
A fellow teacher recently created some terrific materials for this lesson: a graphic organizer and a scaffolded practice packet.
It's not the most meaningful way to factor quadratics, but it works every time. The main reason I like to teach it this way is that even though it's a long process, it's easy to break the learning process up into bite-sized pieces. Finding the greatest common factor of two terms is a skill that can be learned on a different day than factoring simple quadratics with the box, and X marks the spot is just a fun activity on its own. Once everyone is comfortable with all the separate pieces, it's not such a leap to just put them all together.