Solution - Factoring binomials using the difference of squares

Step  1  :

            8
 Simplify   —
            n

Equation at the end of step  1  :

                      8
  ((27•(x3))•(y3))+(((—•u)•l)•l)
                      n
 Step  2  :

Equation at the end of step  2  :

                 8ul2
  (33x3 • y3) +  ————
                  n  

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Adding a fraction to a whole

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

               33x3y3      33x3y3 • n 
     33x3y3 =  ——————  =  ——————————
                 1            n     

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 :

 3.2       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:

 33x3y3 • n + 8ul2      27x3y3n + 8ul2 
 —————————————————  =  ——————————————
         n                   n       

Trying to factor as a Sum of Cubes :

 3.3      Factoring:  27x³y³n + 8ul² 

Theory : A sum of two perfect cubes,  a³ + b³ can be factored into  :
             (a+b) • (a²-ab+b²)
Proof  : (a+b) • (a²-ab+b²) =
    a³-a²b+ab²+ba²-b²a+b³ =
    a³+(a²b-ba²)+(ab²-b²a)+b³=
    a³+0+0+b³=
    a³+b³

Check :  27  is the cube of  3 

Check :  8  is the cube of   2 
Check :  x³ is the cube of   x¹

Check :  y³ is the cube of   y¹

Check :  n ¹ is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes

Final result :

  27x3y3n + 8ul2 
  ——————————————
        n