Solution - Factoring binomials using the difference of squares

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "g2"   was replaced by   "g^2". 

Step  1  :

            t2g3x2 + 1 
 Simplify   ——————————
            tgx + tgxc

Step  2  :

Pulling out like terms :

 2.1     Pull out like factors :

   tgx + tgxc  =   tgx • (c + 1) 

Trying to factor as a Sum of Cubes :

 2.2      Factoring:  t²g³x² + 1 

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 :  1  is the cube of   1 
Check :  t ² is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes

Final result :

    t2g3x2 + 1  
  —————————————
  tgx • (c + 1)