I have seen a lot of methods for factoring a quadratic with a leading coefficient. Out of the ones I've tried, this is my favorite. The analogy to an airplane is a bit of a stretch, but students seem to remember it pretty well. So I'll take it.
I should also mention that, before I show this to students, I always spend some time letting them work on these by trial and error. I figure a process like this is worthless if they don't actually understand what they are doing. Once I feel like students understand the concept but they are still struggling to get every problem to work, I show them this. We treat it like a shortcut, and boy do they appreciate it!
Here is an example:
First, my students know they will need two binomials, so I start with two sets of parenthesis. Then I put the leading coefficient in each parenthesis. Hopefully, the students have a problem with this. We talk about why it is a problem, and I promise them that we will get rid of the extra 2 before we're all done.
Then, multiply a and c. (See the airplane wings? Use your imagination.)
Look for two numbers with product ac and sum b. (Propeller? I know this is really a stretch.)
Put those numbers in the parenthesis.
Divide the extra 2. (The landing? Maybe.) It is pretty cheesy, but when students are having trouble I can say something like "you forgot the landing", and they know what I mean.
Done.
For something like this, you may need to divide both binomials. I point out how dividing by 3 and by 2 is the same as dividing by 6. We just choose the division that will keep integers.
