Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  26a3 -  216

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   64a³ - 216  =   8 • (8a³ - 27) 

Trying to factor as a Difference of Cubes:

 3.2      Factoring:  8a³ - 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 :  8  is the cube of  2 

Check :  27  is the cube of   3 
Check :  a³ is the cube of   a¹

Factorization is :
             (2a - 3)  •  (4a² + 6a + 9) 

Trying to factor by splitting the middle term

 3.3     Factoring  4a² + 6a + 9 

The first term is,  4a²  its coefficient is  4 .
The middle term is,  +6a  its coefficient is  6 .
The last term, "the constant", is  +9 

Step-1 : Multiply the coefficient of the first term by the constant   4 • 9 = 36 

Step-2 : Find two factors of  36  whose sum equals the coefficient of the middle term, which is   6 .

For tidiness, printing of 12 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 :

  8 • (2a - 3) • (4a2 + 6a + 9)