Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  33g3 -  343

Step  2  :

Trying to factor as a Difference of Cubes:

 2.1      Factoring:  27g³-343 

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 :  343  is the cube of   7 
Check :  g³ is the cube of   g¹

Factorization is :
             (3g - 7)  •  (9g² + 21g + 49) 

Trying to factor by splitting the middle term

 2.2     Factoring  9g² + 21g + 49 

The first term is,  9g²  its coefficient is  9 .
The middle term is,  +21g  its coefficient is  21 .
The last term, "the constant", is  +49 

Step-1 : Multiply the coefficient of the first term by the constant   9 • 49 = 441 

Step-2 : Find two factors of  441  whose sum equals the coefficient of the middle term, which is   21 .

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 :

  (3g - 7) • (9g2 + 21g + 49)