Step by step
The question
Question: factor completely x⁴ + 2x³ − 8x².
Completely means: until no part can be factored any more.
1 Common factor first
Every term has x².
So the common factor is x².
We divide every term by it.
What is left goes in the bracket.
So it is x²(x² + 2x − 8).
2 Count the terms
The bracket has 3 terms.
With 3 terms we look for two numbers that multiply to c and add to b.
We need two numbers.
They multiply to −8, and they add to 2.
They multiply to a minus number, so one number is plus and one is minus.
They add to a plus number, so the bigger one is plus.
We list the pairs that make 8, smallest first.
We take the smaller from the bigger.
We need 2.
2 and 4 give 2.
Yes!
With the signs: −2 and 4.
So we get x²(x − 2)(x + 4).
3 Check each bracket
Brackets like x − 2 have just x and a number.
They can't be factored.
No bracket can be factored any more.
So we are done.
Careful!
Don't forget the common factor x² in the answer.
4 Check
Let's check: we multiply it back out.
Each part of one bracket times each part of the other.
Then the common factor times every part.
We get x⁴ + 2x³ − 8x² again.
So it is right!
Let's look back.
First: common factor first.
Then: count the terms.
Then: check each bracket.
Last: check.