Step by step
The question
Question: simplify (x² + 7x + 12)/(x² + 2x − 3).
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 + 7x + 12.
For x² + 7x + 12, we need two numbers.
They multiply to 12, and they add to 7.
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 12, smallest first.
We add each pair.
We need 7.
3 and 4 give 7.
Yes!
So the top is (x + 3)(x + 4).
2 Factor the bottom
We look at the bottom: x² + 2x − 3.
For x² + 2x − 3, we need two numbers.
They multiply to −3, 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 3, smallest first.
We take the smaller from the bigger.
We need 2.
1 and 3 give 2.
Yes!
With the signs: −1 and 3.
So the bottom is (x − 1)(x + 3).
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 + 3)(x + 4)/(x − 1)(x + 3).
(x + 3)/(x + 3) is 1.
Cross them out.
4 Write what is left
What is left is (x + 4)/(x − 1).
One more thing: the bottom of the question can't be 0.
So x can't be −3 or 1.
5 Check
Let's check with a number: x = 2.
First in the question.
We get 6.
Now in our answer.
We get 6 too.
Both are 6.
So (x + 4)/(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.