Solution - Adding, subtracting and finding the least common multiple

Step  1  :

            9x + 8y
 Simplify   ———————
               6   

Equation at the end of step  1  :

       (x+2y)      (9x+8y)
  ((0-(——————•x))+(———————•x))+2
         3            6   

Step  2  :

Equation at the end of step  2  :

       (x+2y)     x•(9x+8y)
  ((0-(——————•x))+—————————)+2
         3            6    

Step  3  :

            x + 2y
 Simplify   ——————
              3   

Equation at the end of step  3  :

       (x+2y)     x•(9x+8y)
  ((0-(——————•x))+—————————)+2
         3            6    

Step  4  :

Equation at the end of step  4  :

      x•(x+2y)  x•(9x+8y)
  ((0-————————)+—————————)+2
         3          6    

Step  5  :

Calculating the Least Common Multiple :

 5.1    Find the Least Common Multiple

      The left denominator is :       3 

      The right denominator is :       6 

      Least Common Multiple:
      6 

Calculating Multipliers :

 5.2    Calculate multipliers for the two fractions


    Denote the Least Common Multiple by  L.C.M 
    Denote the Left Multiplier by  Left_M 
    Denote the Right Multiplier by  Right_M 
    Denote the Left Deniminator by  L_Deno 
    Denote the Right Multiplier by  R_Deno 

   Left_M = L.C.M / L_Deno = 2

   Right_M = L.C.M / R_Deno = 1

Making Equivalent Fractions :

 5.3      Rewrite the two fractions into equivalent fractions

Two fractions are called equivalent if they have the same numeric value.

For example :  1/2   and  2/4  are equivalent,  y/(y+1)²   and  (y²+y)/(y+1)³  are equivalent as well.

To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

   L. Mult. • L. Num.      -x • (x+2y) • 2
   ——————————————————  =   ———————————————
         L.C.M                    6       

   R. Mult. • R. Num.      x • (9x+8y)
   ——————————————————  =   ———————————
         L.C.M                  6     

Adding fractions that have a common denominator :

 5.4       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:

 -x • (x+2y) • 2 + x • (9x+8y)     7x2 + 4xy
 —————————————————————————————  =  —————————
               6                       6    

Equation at the end of step  5  :

  (7x2 + 4xy)    
  ——————————— +  2
       6         

Step  6  :

Rewriting the whole as an Equivalent Fraction :

 6.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  6  as the denominator :

         2     2 • 6
    2 =  —  =  —————
         1       6  

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

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   7x² + 4xy  =   x • (7x + 4y) 

Adding fractions that have a common denominator :

 7.2       Adding up the two equivalent fractions

 x • (7x+4y) + 2 • 6     7x2 + 4xy + 12
 ———————————————————  =  ——————————————
          6                    6       

Trying to factor a multi variable polynomial :

 7.3    Factoring    7x² + 4xy + 12 

Try to factor this multi-variable trinomial using trial and error 

 Factorization fails

Final result :

  7x2 + 4xy + 12
  ——————————————
        6