Ok, I'm not sure if this is any help, but I'll try to run you through how I would solve that one.
6x^2 - 13x + 6
First thing we notice: We have a negative in the second section. And we notice that the last section is positive. That means the 2 last numbers are both negative. Multiply a negative by a negative, and you get a positive. Add a positive * a negative to a positive * a negative, you get a negative, which satisfies the middle.
the 6x^2 can be broken down a number of ways. That means that you may have 6x * 1x for that part of the factored answer. Or you might have 2x * 3x (and they could be in either order).
So now we know that our factored result could be any of:
(1x - ?)(6x - ?)
(6x - ?)(1x - ?)
(2x - ?)(3x - ?)
(3x - ?)(2x - ?)
With a - sign or 2 somewhere in there.
Now we have to think about the -13x. we know that the second section is created by adding the product of the inside numbers to the product of the outside numbers. We have to think, can 13 be made by multiples of 1 and 6? Yes, so can it be made with multiples of 2 and 3? Also yes. Unfortunately, we can't rule out any combination yet, because of this. Which means we have a bit of a guessing game, sorta.
Say you pick a combination at random to examine, and it happens to be (2x + ?)(3x + ?) (the correct choice, for the sake of not taking all day for me to run through everything). We know that 2x * 3x = 6x^2, so that works. Now, here's a bit of a trick for finding the last 2. If you ignore the middle number, the -13x for a second, you'll realize that the last part, +6 must be the product of the 2 missing numbers. So, how many combinations make 6?
We have:
1, 6
6, 1
2, 3
3, 2
But because there is a -13 and a +6, we need to assume that these numbers have to both be negative. So, let's chose the 1, 6, (2x - 1)(3x - 6) and see if that works... but we get -15 instead of -13, so that's a no go. Try it in reverse, (2x -6)(3x - 1) and you get -20 in the middle, even further away.
So we know it's not either 1, 6 or 6, 1. Now we try 2, 3 then 3, 2:
(2x -2)(3x - 3) = 6 x^2-12 x+6
(2x -3)(3x - 2) = 6 x^2-13 x+6
So after trial and error, we know that the answer must be (2x -3)(3x - 2).
This is the slow method for figuring things out, but it works. I forget all the crap you can do to speed things up, because I haven't had to factor for quite some time.
Basically, you figure out all the combinations that could work, and try to rule a bunch of them out early, then do trial and error to figure out which answer fits.
This page seems to be a good read:
http://fp.academic.venturacollege.edu/rbrunner/hints_7.2.htm
