Step by step
The question
Question: factor completely 2x⁴ − 32.
Completely means: until no part can be factored any more.
1 Common factor first
All the numbers can be divided by 2.
So the common factor is 2.
We divide every term by it.
What is left goes in the bracket.
So it is 2(x⁴ − 16).
2 Count the terms
The bracket has 2 terms.
With 2 terms we look for two squares or two cubes.
x⁴ − 16 is one square take away another square.
x⁴ is x² squared.
16 is 4 squared.
A square minus a square is the plus bracket times the minus bracket.
So we get 2(x² + 4)(x² − 4).
3 Factor again
x² − 4 is one square take away another square.
x² is x squared.
4 is 2 squared.
A square minus a square is the plus bracket times the minus bracket.
So we get 2(x² + 4)(x + 2)(x − 2).
4 Check each bracket
Brackets like x + 2 have just x and a number.
They can't be factored.
x² + 4 is a square PLUS a square.
A sum of two squares does not factor.
We call it prime.
No bracket can be factored any more.
So we are done.
Careful!
That is not the end: x² − 4 is two squares again.
5 Check
Let's check: we multiply it back out.
Each part of one bracket times each part of the other.
Then the next bracket.
Then the common factor times every part.
We get 2x⁴ − 32 again.
So it is right!
Let's look back.
First: common factor first.
Then: count the terms.
Then: factor again.
Then: check each bracket.
Last: check.