Solution - Factoring binomials using the difference of squares

Step  1  :

             d
 Simplify   ——
            dx

Canceling Out :

 1.1    Canceling out d as it appears on both sides of the fraction line

Equation at the end of step  1  :

  1
  — • (x3 + 1)7
  x

Step  2  :

Trying to factor as a Sum of Cubes :

 2.1      Factoring:  x³+1 

Put the exponent aside, try to factor  x³+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 :  x³ is the cube of   x¹

Factorization is :
             (x + 1)  •  (x² - x + 1) 

Raise each factor to the exponent which was put aside Factorization becomes :
       (x + 1) ⁷ •  (x² - x + 1) ⁷

Trying to factor by splitting the middle term

 2.2     Factoring  x² - x + 1 

Note, as we are trying to factor ( x² - x + 1)⁷ then, should we find a factorization, we'll raise each of the factors to power 7

The first term is,  x²  its coefficient is  1 .
The middle term is,  -x  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 :

  (x + 1)7 • (x2 - x + 1)7 
  ————————————————————————
             x