Step by step
The question
Question: add 4/x(x + 3) + 1/(x + 3).
1 Find the common bottom
The bottoms are x(x + 3) and (x + 3).
The common bottom must have every factor of both bottoms, each one once.
It is x(x + 3).
Careful!
We can't add the tops and add the bottoms.
Fractions need the same bottom first.
2 Make the bottoms the same
The first fraction already has it.
The second fraction is missing x.
We multiply its top and bottom by x.
x/x is 1, so the value stays the same.
3 Add the tops
Now one fraction: (4 + x)/x(x + 3).
On top: 4 + x is x + 4.
So the top is x + 4.
4 Cross out the same
Is a whole factor on top the same as one on the bottom?
No.
So nothing crosses out.
We leave the bottom as factors.
So the answer is (x + 4)/x(x + 3).
The bottoms can't be 0.
So x can't be −3 or 0.
5 Check
Let's check with a number: x = 1.
First in the question.
4/4 plus 1/4 is 5/4.
Now in our answer.
We get 5/4 too.
Both are 5/4.
So (x + 4)/x(x + 3) is right!
Let's look back.
First: find the common bottom.
Then: make the bottoms the same.
Then: add the tops.
Then: cross out the same.
Last: check.