How to creates Factor by Substitution ?

Can you factor this polynomial?

x² + 3x + 2

(For this tutorial, I'm going to assume that you know how to do some basic factoring.) The polynomial above factors like this:

x² + 3x + 2 = (x + 2)(x + 1).

Now, it factors the same way no matter what the variable is, right?

y² + 3y + 2 = (y + 2)(y + 1).

In fact, you could have any variable, expression or number substituted in place of x and it would still work:

(qr - s )² + 3(qr - s) + 2 = ((qr -s )+2)((qr -s) + 1)

Look! We've managed to factor that really nasty-looking stuff on the left, without much effort at all! Hint: If you're getting confused, it might help to make up a new variable for the expression. For example, if you want to factor.

(x + 1)² - 8(x + 1) + 16,

you can make up a new variable A to represent the expression (x + 1). Then rewrite your problem using the new variable, and you can see that it's easy to factor.

A² - 8A + 16 (Rewrite using the variable A)

(A - 4)² (....it factors as a perfect square.)

Then you can substitute (x + 1) back in place of A.

((x + 1) - 4)²

= (x -3)²