SOLUTION: how can i understand foiling and factoring better?

Question 704297: how can i understand foiling and factoring better?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
Factoring takes practice and it is hard work.
The good news is that factoring gets easier with practice, and it is a very efficient way to solve some quadratic equations.
The bad news is that factoring keeps coming up in math, from rational functions and quadratic equations all the way to calculus.
I will show you my strategies, along with the rationale behind it.
Understanding the rationale helps me remember procedures without having to memorize "recipes" that make no sense to me.
There will be too many words, but I'll use more pictures as it gets too complicated for just words..

I assume you know that the first thing to do when factoring a complicated expression is to look for common factors.
If you have a polynomial like , with in all the terms,
you would first "take out the common factor" , as in

FOIL is an acronym teachers use to help you remember all the terms when multiplying two binomials.
When multiplying the binomials times
you get a sum of all four possible products:

In the acronym FOIL,
F stand for First terms multiplied make
O stands for Outside terms multiplied make
I L stands for stands for Inside terms multiplied make
Last term multiplied make
Thinking of FOIL makes sure you include all four products, just once each, and nothing else.
In , the first and are the Outside terms.
The and the that appear together in the middle are the Inside terms
is the First terms multiplied,
is the Outside terms multiplied,
is the Inside terms multiplied
After the "foiling" often comes the "collecting like terms", so after

you add together the terms in , and to get

and then you have

Factoring is "unfoiling".
You know that the is the product of two first terms that must have been in both binomials.
You know that the term is the product of two last terms, and it is not that clear what those two last terms were.
There are several possibilities (four to be specific):

and
In the foiling, the last terms of the binomial were also used as factors in the Outside and Inside products to make two terms in .
In those two terms in the last terms of the binomial appeared as coefficients in front of x.
The problem is that the two terms in from the foiling were already "collected" together into the term,
so the coefficients were added together to get .
We go back to the four possibilities and see what pair adds up to :

and
so the two last terms must be and .
So the factoring is

A PICTURE:
In the picture below, the large square represents the product .
It is what happens when a patio tiles long by tiles wide
is enlarged by adding rows of tiles to the length and rows to the wdith.
Foiling is multiplying the side measures to get the area of the rectangle.

The figure below illustrates

Factoring is figuring out the rectangle sides from the area.
The figures below illustrates factoring
I will pretend that I do not know what the factors were as I try to undo the multiplication.
If you multiply together opposite corners, you find that they are the same.
and
so the and that we are looking for must be factors of

We look for pairs of factors that multiply to without worrying much about signs, or about the .
I know that 1,2,3,4,5,6,8,10,and 12 are factors because they divide 120 evenly).
Dividing 120 by each of the small,easy to find factors, I find the matching larger factors.

When I get to I realize that involves the same pair of factors I had already found as , so I know I have found all the factors.
Because the product has a minus sign, I know that one of the factors must have a minus sign.
Because they must add to the in , I figure that I am looking for a pair of factors that differ by and I am going to give the minus sign to the larger factor.
is the answer
does not work, and neither do the other pairs.
and are the coefficients I am looking for.
I fill the and squares with and .
It does not matter where I put each one.

Now I need to figure common factors for each row and column. Those common factors will replace the question marks.
I start with what seems more obvious to me, whatever I can figure out faster.
The and in the left column share as a common factor, so I put it above the left column.
The and in the top row share as a common factor, so I put it to the left of the top row. I could also have figured it, from what I already knew, as
.
The other question marks, I can figure out from what I already know:
goes above and
goes above .

SPECIAL PRODUCTS:
There are special products that you end up remembering after a while, and are represented by the "formulas" below.
Square of a binomial:

Difference of squares:

That one is easy to explain because when you FOIL, the O and I products cancel out, so your FOIL turns into FL (I cal that Florida).
Difference of cubes:

Sum of cubes:

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