Solution - Factoring multivariable polynomials

Step  1  :

Equation at the end of step  1  :

    (((6•(x2))-7xy)-(3•(y2)))
  ((—————————————————————————•(x2))-5xy)+3y2
                2            
 Step  2  :

Equation at the end of step  2  :

    (((6•(x2))-7xy)-3y2)
  ((————————————————————•x2)-5xy)+3y2
             2          
 Step  3  :

Equation at the end of step  3  :

    (((2•3x2)-7xy)-3y2)
  ((———————————————————•x2)-5xy)+3y2
             2         

Step  4  :

            6x2 - 7xy - 3y2
 Simplify   ———————————————
                   2       

Trying to factor a multi variable polynomial :

 4.1    Factoring    6x² - 7xy - 3y² 

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

 Found a factorization  :  (2x - 3y)•(3x + y)

Equation at the end of step  4  :

    (2x - 3y) • (3x + y)                   
  ((———————————————————— • x2) -  5xy) +  3y2
             2                             

Step  5  :

Equation at the end of step  5  :

   x2 • (2x - 3y) • (3x + y)            
  (————————————————————————— -  5xy) +  3y2
               2                        

Step  6  :

Rewriting the whole as an Equivalent Fraction :

 6.1   Subtracting a whole from a fraction

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

           5xy     5xy • 2
    5xy =  ———  =  ———————
            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

Adding fractions that have a common denominator :

 6.2       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:

 x2 • (2x-3y) • (3x+y) - (5xy • 2)     6x4 - 7x3y - 3x2y2 - 10xy 
 —————————————————————————————————  =  —————————————————————————
                 2                                 2            

Equation at the end of step  6  :

  (6x4 - 7x3y - 3x2y2 - 10xy)     
  ——————————————————————————— +  3y2
               2                 

Step  7  :

Rewriting the whole as an Equivalent Fraction :

 7.1   Adding a whole to a fraction

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

           3y2     3y2 • 2
    3y2 =  ———  =  ———————
            1         2   

Step  8  :

Pulling out like terms :

 8.1     Pull out like factors :

   6x⁴ - 7x³y - 3x²y² - 10xy  = 

  x • (6x³ - 7x²y - 3xy² - 10y) 

Checking for a perfect cube :

 8.2    6x³ - 7x²y - 3xy² - 10y  is not a perfect cube

Adding fractions that have a common denominator :

 8.3       Adding up the two equivalent fractions

 x • (6x3-7x2y-3xy2-10y) + 3y2 • 2      6x4 - 7x3y - 3x2y2 - 10xy + 6y2 
 —————————————————————————————————  =  ———————————————————————————————
                 2                                    2               

Final result :

  6x4 - 7x3y - 3x2y2 - 10xy + 6y2 
  ———————————————————————————————
                 2