Solution - Factoring multivariable polynomials

Step  1  :

Equation at the end of step  1  :

  (5 • (a3)) -  (23•33•5p3)
 Step  2  :

Equation at the end of step  2  :

  5a3 -  (23•33•5p3)

Step  3  :

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   5a³ - 1080p³  =   5 • (a³ - 216p³) 

Trying to factor as a Difference of Cubes:

 4.2      Factoring:  a³ - 216p³ 

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 :  216  is the cube of   6 
Check :  a³ is the cube of   a¹

Check :  p³ is the cube of   p¹

Factorization is :
             (a - 6p)  •  (a² + 6ap + 36p²) 

Trying to factor a multi variable polynomial :

 4.3    Factoring    a² + 6ap + 36p² 

Try to factor this multi-variable trinomial using trial and error 

 Factorization fails

Final result :

  5 • (a - 6p) • (a2 + 6ap + 36p2)