Solution - Simplification or other simple results

Step  1  :

Trying to factor as a Difference of Squares :

 1.1      Factoring:  s⁵⁰⁴-27 

Theory : A difference of two perfect squares,  A² - B²  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A² - AB + BA - B² =
         A² - AB + AB - B² =
         A² - B²

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 27 is not a square !!

Ruling : Binomial can not be factored as the difference of two perfect squares.

Trying to factor as a Difference of Cubes:

 1.2      Factoring:  s⁵⁰⁴-27 

Theory : A difference 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 :  s⁵⁰⁴ is the cube of   s¹⁶⁸

Factorization is :
             (s¹⁶⁸ - 3)  •  (s³³⁶ + 3s¹⁶⁸ + 9) 

Trying to factor as a Difference of Squares :

 1.3      Factoring:  s¹⁶⁸ - 3 

Check : 3 is not a square !!

Ruling : Binomial can not be factored as the difference of two perfect squares.

Trying to factor as a Difference of Cubes:

 1.4      Factoring:  s¹⁶⁸ - 3 

Check :  3  is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes

Trying to factor by splitting the middle term

 1.5     Factoring  3s¹⁶⁸ + 9 + s³³⁶ 

The first term is,  3s¹⁶⁸  its coefficient is  1 .
The middle term is,  +9  its coefficient is  3 .
The last term, "the constant", is  +s³³⁶ 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 9 = 9 

Step-2 : Find two factors of  9  whose sum equals the coefficient of the middle term, which is   3 .

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Final result :

  (s168 - 3) • (3s168 + 9 + s336)