Let's try another example which requires factoring in steps 1 and 2:
5x3 - 10x2 - 15x
Again, the three steps in Factoring Completely are:
We see that the terms in our example have a greatest common factor of 5x. As instructed, we will factor out this GCF:
5x(x2 - 2x - 3)
We see that (x2 - 2x - 3) is a factorable trinomial, so we factor it:
5x(x + 1)(x - 3)
Proceeding to Step 3, we can look over our expression and see that neither 5x, nor (x + 1), nor (x - 3) can be factored as a difference between two squares. We have factored 5x3 - 10x2 - 15x completely.
For our final example, we will make use of all three Factoring Completely steps.
12x4 - 3x2 - 54
We factor out a Greatest Common Factor of 3.
3(4x4 - x2 - 18)
3(4x2 - 9)(x2 + 2)
Finally, we identify (4x2 - 9) as a binomial that can be factored into (2x + 3)(2x - 3). So the completely factored result is
3(2x + 3)(2x - 3)(x2 + 2).