Step by step
The question
Question: add 3/x(x + 1) + 2 over x.
1 Find the common bottom
The bottoms are x(x + 1) and x.
The common bottom must have every factor of both bottoms, each one once.
It is x(x + 1).
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 + 1).
We multiply its top and bottom by (x + 1).
(x + 1)/(x + 1) is 1, so the value stays the same.
3 Add the tops
Now one fraction: (3 + 2(x + 1)) over x(x + 1).
4 Open the brackets
The 2 in front of the bracket multiplies every part inside.
So we write 2 times each part.
2 times x is 2x.
2 times 1 is 2.
5 Group like terms
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
Plain numbers go together.
Each colour is one kind.
We move like terms next to each other.
Each term takes its sign with it.
3 + 2 is 5.
So the expression is 2x + 5.
6 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 (2x + 5)/x(x + 1).
The bottoms can't be 0.
So x can't be −1 or 0.
7 Check
Let's check with a number: x = 1.
First in the question.
3/2 plus 2 is 7/2.
Now in our answer.
We get 7/2 too.
Both are 7/2.
So (2x + 5)/x(x + 1) is right!
Let's look back.
First: find the common bottom.
Then: make the bottoms the same.
Then: add the tops.
Then: open the brackets.
Then: group like terms.
Then: cross out the same.
Last: check.