Solution - Reducing fractions to their lowest terms

Step  1  :

            y3
 Simplify   ——
            2 

Equation at the end of step  1  :

                      y3
  (((4•(x2))-2xy)-((3•——)•x))+3y
                      2 
 Step  2  :

Equation at the end of step  2  :

                    3xy3     
  ((22x2 -  2xy) -  ————) +  3y
                     2       

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  2  as the denominator :

                  4x2 - 2xy     (4x2 - 2xy) • 2
     4x2 - 2xy =  —————————  =  ———————————————
                      1                2       

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  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   4x² - 2xy  =   2x • (2x - y) 

Adding fractions that have a common denominator :

 4.2       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:

 2x • (2x-y) • 2 - (3xy3)     8x2 - 3xy3 - 4xy
 ————————————————————————  =  ————————————————
            2                        2        

Equation at the end of step  4  :

  (8x2 - 3xy3 - 4xy)    
  —————————————————— +  3y
          2             

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  2  as the denominator :

          3y     3y • 2
    3y =  ——  =  ——————
          1        2   

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   8x² - 3xy³ - 4xy  =   x • (8x - 3y³ - 4y) 

Trying to factor a multi variable polynomial :

 6.2    Factoring    8x - 3y³ - 4y 

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

 Factorization fails

Adding fractions that have a common denominator :

 6.3       Adding up the two equivalent fractions

 x • (8x-3y3-4y) + 3y • 2     8x2 - 3xy3 - 4xy + 6y
 ————————————————————————  =  —————————————————————
            2                           2          

Checking for a perfect cube :

 6.4    8x² + 3xy³ + 4xy + 6y  is not a perfect cube

Final result :

 8x2 + 3xy3 + 4xy + 6y