Solution - Adding, subtracting and finding the least common multiple

Step  1  :

             y
 Simplify   ——
            x2

Equation at the end of step  1  :

     x    y           y
  (((—-y)-—)+y)-((2x•——)•y2)
     x    x          x2
 Step  2  :

Multiplying exponential expressions :

 2.1    y¹ multiplied by y² = y^{(1 + 2)} = y³

Equation at the end of step  2  :

     x    y     2y3
  (((—-y)-—)+y)-———
     x    x      x 

Step  3  :

            y
 Simplify   —
            x

Equation at the end of step  3  :

     x          y           2y3
  (((— -  y) -  —) +  y) -  ———
     x          x            x 

Step  4  :

            x
 Simplify   —
            x

Equation at the end of step  4  :

                y           2y3
  (((1 -  y) -  —) +  y) -  ———
                x            x 

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Subtracting a fraction from a whole

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

              1 - y     (1 - y) • x
     1 - y =  —————  =  ———————————
                1            x     

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 :

 5.2       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:

 (1-y) • x - (y)     -xy + x - y
 ———————————————  =  ———————————
        x                 x     

Equation at the end of step  5  :

   (-xy + x - y)          2y3
  (————————————— +  y) -  ———
         x                 x 

Step  6  :

Rewriting the whole as an Equivalent Fraction :

 6.1   Adding a whole to a fraction

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

         y     y • x
    y =  —  =  —————
         1       x  

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   -xy + x - y  =   -1 • (xy - x + y) 

Adding fractions that have a common denominator :

 7.2       Adding up the two equivalent fractions

 (-xy+x-y) + y • x     x - y
 —————————————————  =  —————
         x               x  

Equation at the end of step  7  :

  (x - y)    2y3
  ——————— -  ———
     x        x 

Step  8  :

Adding fractions which have a common denominator :

 8.1       Adding fractions which 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-y) - (2y3)     x - 2y3 - y
 —————————————  =  ———————————
       x                x     

Trying to factor a multi variable polynomial :

 8.2    Factoring    x - 2y³ - y 

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

 Factorization fails

Final result :

  x + 2y3 + y
  ———————————
       x