Multiplying Rational Expressions  Step by Step 
Rational expressions are multiplied in the same way as you would with regular fractions. As you may have learned, we multiply simple fractions using the steps below...


Steps in Multiplying Rational Expressions
 Multiply the numerators together
 Multiply the denominators together
 Simplify the "new" fraction by cancelling common factors when possible

Let us proceed by going over five (5) worked examples in this lesson.
Direction: Simplify the given rational expressions by multiplication.
Example 1: Multiply the rational expressions .
I see that both denominators are factorable. The first denominator is a case of difference of two squares. The second denominator is easy because I can pull out a factor of x.
This is how it looks.


Factor the denominators 


Now, I can multiply across the numerators and across the denominators by placing them side by side.
At this point, I compare the top and bottom factors and decide which ones can be crossedout. 


By color coding the common factors, it is clear which ones to eliminate.
The factors (2x+1) and (x+1) should both be cancelled out as shown. 


What remains on top is just the number 1.
And so we have this as our final answer.
Try not to distribute it back and keep it in factored form. However, if your teacher wants the final answer to be distributed, then do so. Either case should be correct. It's just a matter of preference. 
Example 2: Multiply the rational expressions .
We need to factor out all the trinomials. The good news is that this type of trinomial, where the coefficient of the squared term is 1, is very easy to handle. Starting with the first numerator, find two numbers where its product gives the last term (10) and its sum gives the middle coefficient (7). I'm thinking of +5 and +2. They should work, agree? For the second numerator, the two numbers must be −7 and +1 since their product is the last term (−7) while its sum is the middle coefficient (−6).
To factor the first denominator, find two numbers with a product of the last term (14) and a sum of the middle coefficient (9). By trial and error, the numbers are −2 and −7. However, you should always verify it. Now for the second denominator, think of two numbers when multiplied gives the last term (5) and when added gives (6). Obviously they are +5 and +1.
The correct factors for each trinomial should appear like this.


Factor the numerators and denominators completely. 


Multiply them together  numerator against numerator and denominator against denominator.
In other words, place side by side in a single fractional symbol. 


The color schemes should aid in identifying common factors that we can get rid of. 


Yep! We cleaned it out beautifully. This is our final answer. 
Example 3: Multiply the rational expressions .
In this problem, I have six terms that need factoring. However, most of them are easy to handle and I will provide suggestions on how to factor each. I will also show a quick side calculation how to factor 4x^{2}+x3 because it can be challenging to some.
Example 4: Multiply the rational expressions .
As you can see, there are so many things going on in this problem. However, don't be intimidated by how it looks. Start by factoring each term completely. The problem will become easier as you go along.
In fact, once we have factored the terms correctly, the rest of the steps becomes manageable.
Example 5: Multiply the rational expressions .
I am sure that by now, you are getting better on how to factor. The only thing I need to point out is the denominator of the first rational expression, x^{3}−1. This is a special case called difference of two cubes.
In this problem, I will use Case 2 because of the "minus"; hence the word difference. Case 1 is known as the sum of two cubes because of the "plus" symbol.
These are the factors...

