My teacher just said it like this:
Take the example of:
Now, is the "target number"
We need to split into it's factors,
and try to find a combination that will make 6. This is , so the factors are
The next example he explained were of the type , ie. because these factored into things of the form so they were easy to get our head around. Now, is our target number and we look at the factors of :
and we can get the target number with , so the factors are
Then came , ie . Again, is our "target number" and again we look at the factors of which are and
clearly so the factors are and then we could just adjust the signs as necessary to get .
When it came to things of the form , it would be the same thing, except this time you'd have to use both the factors of and the factors of to get the target number.
For cubics, quartics, etc, which we didn't cover until 2 years later, we just learned the factor, remainder and rational roots theorems, which made things very straightforward.