Solution - Reducing fractions to their lowest terms

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "y2"   was replaced by   "y^2".  3 more similar replacement(s).

Step  1  :

Equation at the end of step  1  :

               (y2)
  ((((x2)-2xy)-————)-4xy)+3y2
               (x2)
 Step  2  :
            y2
 Simplify   ——
            x2

Equation at the end of step  2  :

                     y2             
  ((((x2) -  2xy) -  ——) -  4xy) +  3y2
                     x2             

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  x²  as the denominator :

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

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 :

   x² - 2xy  =   x • (x - 2y) 

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:

 x • (x-2y) • x2 - (y2)     x4 - 2x3y - y2
 ——————————————————————  =  ——————————————
           x2                     x2      

Equation at the end of step  4  :

   (x4 - 2x3y - y2)            
  (———————————————— -  4xy) +  3y2
          x2                   

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  x²  as the denominator :

           4xy     4xy • x2
    4xy =  ———  =  ————————
            1         x2   

Trying to factor a multi variable polynomial :

 5.2    Factoring    x⁴ - 2x³y - y² 

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

 Factorization fails

Adding fractions that have a common denominator :

 5.3       Adding up the two equivalent fractions

 (x4-2x3y-y2) - (4xy • x2)      x4 - 6x3y - y2
 —————————————————————————  =  ——————————————
            x2                       x2      

Equation at the end of step  5  :

  (x4 - 6x3y - y2)    
  ———————————————— +  3y2
         x2           

Step  6  :

Rewriting the whole as an Equivalent Fraction :

 6.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  x²  as the denominator :

           3y2     3y2 • x2
    3y2 =  ———  =  ————————
            1         x2   

Trying to factor a multi variable polynomial :

 6.2    Factoring    x⁴ - 6x³y - y² 

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

 (x4-6x3y-y2) + 3y2 • x2      x4 - 6x3y + 3x2y2 - y2 
 ———————————————————————  =  ——————————————————————
           x2                          x2          

Checking for a perfect cube :

 6.4    x⁴ + 6x³y + 3x²y² + y²  is not a perfect cube

Final result :

 x4 + 6x3y + 3x2y2 + y2