Step by step
The question
Question: simplify (x² + 5x + 6)/(x + 2)(x − 1).
A fraction with x in it is called a rational expression.
We make it simpler like a number fraction.
6/9: both have a 3 in them.
3/3 is 1, so 2/3 is left.
We do the same with brackets.
1 Factor the top
We look at the top: x² + 5x + 6.
For x² + 5x + 6, we need two numbers.
They multiply to 6, and they add to 5.
They multiply to a plus number, so the two numbers have the same sign.
They add to a plus number, so both are plus.
We list the pairs that make 6, smallest first.
We add each pair.
We need 5.
2 and 3 give 5.
Yes!
So the top is (x + 2)(x + 3).
2 Factor the bottom
We look at the bottom: (x + 2)(x − 1).
It is already factored: (x + 2)(x − 1) means (x + 2) times (x − 1).
We just read the factors.
So the bottom is (x + 2)(x − 1).
3 Cross out the same
Careful!
We cross out only a whole bracket, or a whole x, that multiplies everything.
Never one part of a bracket.
Now we write the fraction with the factors: (x + 2)(x + 3)/(x + 2)(x − 1).
(x + 2)/(x + 2) is 1.
Cross them out.
4 Write what is left
What is left is (x + 3)/(x − 1).
One more thing: the bottom of the question can't be 0.
So x can't be −2 or 1.
5 Check
Let's check with a number: x = 2.
First in the question.
We get 5.
Now in our answer.
We get 5 too.
Both are 5.
So (x + 3)/(x − 1) is right!
Let's look back.
First: factor the top.
Then: factor the bottom.
Then: cross out the same.
Then: write what is left.
Last: check.