Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  (((6•(n2))+30n)-36)
  ——————————————————— ÷ (((2•5n2)+15n)-25)
  (((3•(n2))+10n)-48)
 Step  2  :

Equation at the end of step  2  :

  (((6•(n2))+30n)-36)
  ——————————————————— ÷ (10n2+15n-25)
    ((3n2+10n)-48)   
 Step  3  :

Equation at the end of step  3  :

  (((2•3n2)+30n)-36)
  —————————————————— ÷ (10n2+15n-25)
     (3n2+10n-48)   

Step  4  :

            6n2 + 30n - 36
 Simplify   ——————————————
            3n2 + 10n - 48

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   6n² + 30n - 36  =   6 • (n² + 5n - 6) 

Trying to factor by splitting the middle term

 5.2     Factoring  n² + 5n - 6 

The first term is,  n²  its coefficient is  1 .
The middle term is,  +5n  its coefficient is  5 .
The last term, "the constant", is  -6 

Step-1 : Multiply the coefficient of the first term by the constant   1 • -6 = -6 

Step-2 : Find two factors of  -6  whose sum equals the coefficient of the middle term, which is   5 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -1  and  6 
                     n² - 1n + 6n - 6

Step-4 : Add up the first 2 terms, pulling out like factors :
                    n • (n-1)
              Add up the last 2 terms, pulling out common factors :
                    6 • (n-1)
Step-5 : Add up the four terms of step 4 :
                    (n+6)  •  (n-1)
             Which is the desired factorization

Trying to factor by splitting the middle term

 5.3     Factoring  3n²+10n-48 

The first term is,  3n²  its coefficient is  3 .
The middle term is,  +10n  its coefficient is  10 .
The last term, "the constant", is  -48 

Step-1 : Multiply the coefficient of the first term by the constant   3 • -48 = -144 

Step-2 : Find two factors of  -144  whose sum equals the coefficient of the middle term, which is   10 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -8  and  18 
                     3n² - 8n + 18n - 48

Step-4 : Add up the first 2 terms, pulling out like factors :
                    n • (3n-8)
              Add up the last 2 terms, pulling out common factors :
                    6 • (3n-8)
Step-5 : Add up the four terms of step 4 :
                    (n+6)  •  (3n-8)
             Which is the desired factorization

Canceling Out :

 5.4    Cancel out  (n+6)  which appears on both sides of the fraction line.

Equation at the end of step  5  :

  6 • (n - 1)
  ——————————— ÷ (10n2 + 15n - 25)
    3n - 8   

Step  6  :

         6•(n-1)      
 Divide  ———————  by  10n2+15n-25
         (3n-8)       

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   10n² + 15n - 25  =   5 • (2n² + 3n - 5) 

Trying to factor by splitting the middle term

 7.2     Factoring  2n² + 3n - 5 

The first term is,  2n²  its coefficient is  2 .
The middle term is,  +3n  its coefficient is  3 .
The last term, "the constant", is  -5 

Step-1 : Multiply the coefficient of the first term by the constant   2 • -5 = -10 

Step-2 : Find two factors of  -10  whose sum equals the coefficient of the middle term, which is   3 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -2  and  5 
                     2n² - 2n + 5n - 5

Step-4 : Add up the first 2 terms, pulling out like factors :
                    2n • (n-1)
              Add up the last 2 terms, pulling out common factors :
                    5 • (n-1)
Step-5 : Add up the four terms of step 4 :
                    (2n+5)  •  (n-1)
             Which is the desired factorization

Canceling Out :

 7.3    Cancel out  (n-1)  which appears on both sides of the fraction line.

Final result :

             6           
  ———————————————————————
  5 • (3n - 8) • (2n + 5)