Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  ((12 • (x25)) • x) -  7x2
 Step  2  :

Equation at the end of step  2  :

  ((22•3x25) • x) -  7x2
 Step  3  :
 Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   12x²⁶ - 7x²  =   x² • (12x²⁴ - 7) 

Trying to factor as a Difference of Squares :

 4.2      Factoring:  12x²⁴ - 7 

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 :  12  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:

 4.3      Factoring:  12x²⁴ - 7 

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 :  12  is not a cube !!

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

Final result :

  x2 • (12x24 - 7)