Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  
 Step  2  :

Equation at the end of step  2  :

Step  3  :

            4x2 + 4x
 Simplify   ————————
            2x2 - 2 

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   4x² + 4x  =   4x • (x + 1) 

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   2x² - 2  =   2 • (x² - 1) 

Trying to factor as a Difference of Squares :

 5.2      Factoring:  x² - 1 

Theory : A difference of two perfect squares,  A² - B²  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A² - AB + BA - B² =
         A² - AB + AB - B² =
         A² - B²

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 1 is the square of 1
Check :  x²  is the square of  x¹ 

Factorization is :       (x + 1)  •  (x - 1) 

Canceling Out :

 5.3    Cancel out  (x + 1)  which appears on both sides of the fraction line.

Final result :

    2x 
  —————
  x - 1