How do you do FOIL (as in math) backwards

Like say if you had ax^2+bx+c and you had to turn it into a factored form how exactly would you do that? Provide stepbystep example and you'll get best answer! And answer quickly cause I should know this already!

Step 1. Get *all* the factors for a and for c
Step 2 - either by trial and error, or by your cleverness, determine which combination of factors might give you 'b'

In step one, when I am tutoring this, I show the kids I work with how to factor almost any number(at least in high school basic algebra) by knowing the primes (2, 3, 5, 7, 11...)
Normally, you rarely need to know any more primes than to 11 (in high school, that is...)

Assume a = 9, b = -35, c = 24
factors for a are (1,9) and (3,3) (as well as (-1,-9), (-3,-3) )
factors for c are (1,24), (2,12), (3,8), (4, 6) (and all the negative numbers of the same)

By careful examination, I can see that (-3 x 9) + (-8 x 1) = -27 + -8 = -35 (b)
so the factors of a are 1, 9 and the factors of c are -3 & -8

(1x - 3)*(9x - 8)

what i do is called the diamond. it is sort of hard to explain but what you do is make an x.
in the top quadrant put a muliplied by c...usually a is one so it usually is just c.
in the lower quadrant put b.
now for the sides find two numbers that when multiplied equal the tope and when added together equal the bottome.
heres a easy example:

x^2+8x+12
step one make an x like this \ a(c) /
\ /
\ /
/ \
/ \
/ b \
replace the letters with numbers
\ 1(12) /
\ /
\ /