Step by step
The question
Question: simplify (x² + 5x + 6)/(x² − 4) × (x − 2)/(x + 3).
1 Factor every part
First the top of the first fraction: 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 of the first fraction is (x + 2)(x + 3).
Now the bottom of the first fraction: x² − 4.
x² − 4 is x² minus 2 squared.
A square minus a square is the plus bracket times the minus bracket.
So the bottom of the first fraction is (x + 2)(x − 2).
2 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 every part is factored.
A bracket on any top and the same bracket on any bottom make 1.
(x + 2)/(x + 2) is 1.
Cross them out.
(x + 3)/(x + 3) is 1.
Cross them out.
(x − 2)/(x − 2) is 1.
Cross them out.
3 Multiply what is left
Top times top, bottom times bottom.
What is left is 1.
One more thing: the bottom of the question can't be 0.
So x can't be −3, −2 or 2.
4 Check
Let's check with a number: x = 1.
First in the question.
We get 1.
Now in our answer.
We get 1 too.
Both are 1.
So 1 is right!
Let's look back.
First: factor every part.
Then: cross out the same.
Then: multiply what is left.
Last: check.