Solution - Adding, subtracting and finding the least common multiple

Step  1  :

               2  
 Simplify   ——————
            n2 - 1

Trying to factor as a Difference of Squares :

 1.1      Factoring:  n² - 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 :  n²  is the square of  n¹ 

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

Equation at the end of step  1  :

                           2        
  (mn + m - 2n) -  —————————————————
                   (n + 1) • (n - 1)

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  (n+1) • (n-1)  as the denominator :

                    mn + m - 2n     (mn + m - 2n) • (n + 1) • (n - 1)
     mn + m - 2n =  ———————————  =  —————————————————————————————————
                         1                  (n + 1) • (n - 1)        

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 2.2       Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 (mn+m-2n) • (n+1) • (n-1) - (2)     mn3 + mn2 - mn - m - 2n3 + 2n - 2 
 ———————————————————————————————  =  —————————————————————————————————
        1 • (n+1) • (n-1)                  1 • (n + 1) • (n - 1)      

Final result :

  mn3 + mn2 - mn - m - 2n3 + 2n - 2 
  —————————————————————————————————
          (n + 1) • (n - 1)