Solution - Factoring binomials using the difference of squares

Step  1  :

Trying to factor as a Sum of Cubes :

 1.1      Factoring:  x³+125 

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 :  125  is the cube of   5 
Check :  x³ is the cube of   x¹

Factorization is :
             (x + 5)  •  (x² - 5x + 25) 

Trying to factor by splitting the middle term

 1.2     Factoring  x² - 5x + 25 

The first term is,  x²  its coefficient is  1 .
The middle term is,  -5x  its coefficient is  -5 .
The last term, "the constant", is  +25 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 25 = 25 

Step-2 : Find two factors of  25  whose sum equals the coefficient of the middle term, which is   -5 .

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

Final result :

  (x + 5) • (x2 - 5x + 25)