Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  23x3 -  729

Step  2  :

Trying to factor as a Difference of Cubes:

 2.1      Factoring:  8x³-729 

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 :  8  is the cube of  2 

Check :  729  is the cube of   9 
Check :  x³ is the cube of   x¹

Factorization is :
             (2x - 9)  •  (4x² + 18x + 81) 

Trying to factor by splitting the middle term

 2.2     Factoring  4x² + 18x + 81 

The first term is,  4x²  its coefficient is  4 .
The middle term is,  +18x  its coefficient is  18 .
The last term, "the constant", is  +81 

Step-1 : Multiply the coefficient of the first term by the constant   4 • 81 = 324 

Step-2 : Find two factors of  324  whose sum equals the coefficient of the middle term, which is   18 .

For tidiness, printing of 24 lines which failed to find two such factors, was suppressed

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

Final result :

  (2x - 9) • (4x2 + 18x + 81)