Solution - Simplification or other simple results

Step  1  :

Step  2  :

Pulling out like terms :

 2.1     Pull out like factors :

   a⁵ - a²  =   a² • (a³ - 1) 

Trying to factor as a Difference of Cubes:

 2.2      Factoring:  a³ - 1 

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 :  1  is the cube of   1 
Check :  a³ is the cube of   a¹

Factorization is :
             (a - 1)  •  (a² + a + 1) 

Trying to factor by splitting the middle term

 2.3     Factoring  a² + a + 1 

The first term is,  a²  its coefficient is  1 .
The middle term is,  +a  its coefficient is  1 .
The last term, "the constant", is  +1 

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

Step-2 : Find two factors of  1  whose sum equals the coefficient of the middle term, which is   1 .

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Final result :

  a2 • (a - 1) • (a2 + a + 1)