Hi SlowlyFading!

Here's another way of looking at the factoring of ax^2 + bx + c  ( a not zero) trinomials.  It is fully
equivalent to the FOIL method, but has a two dimensional set up rather than one dimensional.

The FOIL technique is equivalent to RCM (reversing cross multiplication for a 2x2 multiplication).

Here is the pattern:

.  .    .  .    .  .   where these three patterns from RIGHT TO LEFT produce
|        X        |   the units, 10', and 100's columns of a product in base 10.
.  .    .  .    .  .   In an unknown base x (polynomials) this is units, x's, x^2's.

Example:

        2  -3   Obtain the 3*(-4)=-12 from the RIGHT HAND pattern.
                  Obtain  2*4+3*(-3) = -1  from  the CROSS pattern.
        3   4   Obtain 2*3=6 from the LEFT HAND pattern.
        ____

   6  -1 -12   

As polynomials this is equivalent to (2x-3)(3x+4) = 6x^2 - x - 12.

Putting this "in reverse" we would be given the 6  -1  -12 and asked to find  the factors of this
which are  2  -3   and  3   4.

Ignoring the signs temporarily there are 6 possible ways to pair the factorizations ( 1*6 or 2*3 )
with the factorizations  ( 1*12 or 2*6 or 3*4 ).  Then we can switch the numbers in one of the
columns (but not in both at the same time)  which doubles the number of possibilities to 12. 

It's a matter of trial and error to find out which combination works (if any).  The more practice
one has doing this, the faster they can find the right one.

HELPFUL HINT for eliminating some possibilities:  If the product ( in this case the 6  -1  -12 ) has no
common factor > 1 (as is the case here) then NEITHER of the two factors we are looking for can
have a common factor > 1.  This eliminates lots of possibilities.  For example:  2   4  or  2   6
or  2   2  or  2  12  or  3   6  or  3   12  or   6   2   or   6    3   or   6   4   or   6   12  (ignoring
signs temporarily).   That's 10 of the 12 possibilities that can be eliminated without even multiplying
the two factors out.  All we have left is  (1   12  with  6   1) or (2   3  with   3   4).

Then since the rightmost term -12 in the product is negative we must make one of the numbers in
the rightmost column negative.  (If the leftmost column of the answer is positive (here 6) then it is
customary to use only positive numbers in the leftmost column although we could make them both
negative.)

The 1  12  with  6   1 would produce a middle term of  71 or -71 (considering the signs) and
the 2   3   with  3   4  would produce a middle term of 1 or -1.  The -1 is what we need.

So  (2  -3)*(3   4) produces the 6   -1   -12  as desired; that is, (2x-3)(3x+4)=6x^2 - x - 12.

My high school algebra book (from the 1940's) taught factoring trinomials of this type this way
instead of the FOIL setup.  I like it better and find that most of my students I show it to like it better also.

But it's a "take it or leave it" proposition.  Choose the one you like best. 

Hi Slowly Fading!

Have you access to an explanation of the "FOIL" method?  It can be used to do the factoring of #3 through #9.  (10 doesn't factor).  This method will work on perfect square trinomials such as
9x^2 - 24x + 16 = (3x-4)^2; and on trial and error types such as x^2 - 2x - 24 = (x-6)(x+4); and on
differences of squares such as 9x^2 - 4 = 9x^2 + 0x - 4 = (3x-2)(3x+2).

Let us know how it's going.

Have a great day!

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