Aim: How do we multiply and divide rational expressions ?

Multiplying and dividing may seem difficult, but when using them to solve rational expressions, it can be a lot easier. First, we start out by factoring out all of the numerators and the denominators so we can make things much easier for us. Then we get to cancel out all of the common factors to simplify the expressions. Lastly we have to multiply the remaining factors in the numerator & the denominator. We can find out the answer with these simple steps.
   Problem:

2x +10x + 12x          x + 7 
____________  *     _____      
    
    2x + 10x               x² − 9

                        ↓

2x(x +5x +6)       x + 7 
__________  *    ________            (Factor it Out)
2x(x + 5)          (x+3)(x−3)

                   ↓

 2x(x+2)(x+3)           x + 7 
____________   *    _______             (Cross Out)
2x(x + 5)              (x+3)(x−3)
                   ↓

 (x+2)          (x +7)         x² +9x + 14    (Multiply Remaining Factors)
______   *   ______  =   ___________

 (x+5)          (x−3)          x² +2x − 15

  
                   ↓

x² +9x + 14
__________                (x cannot equal; 0, 3,−3,−5)
x²  +2x − 15


DIVIDING:
   First you start out by using KCF (Keep, Change, Flip) which is the multiplication of reciprocal.
  Problem : 
x + 7                   x² + 9x + 14                        x + 7            3x² - 9x
____        /      ___________        KCF        _____    *   __________
x² − 9                  3x² − 9x                             x² − 9       x² + 9x + 14

Step 2: Factor out everything that needs to be factored.
     x + 7                 3x(x − 3)
_________     *     ________
(x− 3)(x+3)          (x+7)(x+2)

Step 3: Cancel out the common factors

    x + 7                 3x(x − 3)
_________     *     ________
(x− 3)(x+3)          (x+7)(x+2)

Step 4: Multiply the remaining factors of the numerator & denominator.

      3x                        3x
_________    *    ____________
(x+2)(x+3)           x² + 5X + 6

Final Answer

    3x
__________     (x cannot equal;  0,−2,3,−7,−3)
x² + 5X + 6


http://www.mathwarehouse.com/algebra/rational-expression/how-to-multiply-divide-rational-expressions.php