Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  ((2x2 -  x) -  1) • (x26 - 6)

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  2x²-x-1 

The first term is,  2x²  its coefficient is  2 .
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   2 • -1 = -2 

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

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -2  and  1 
                     2x² - 2x + 1x - 1

Step-4 : Add up the first 2 terms, pulling out like factors :
                    2x • (x-1)
              Add up the last 2 terms, pulling out common factors :
                     1 • (x-1)
Step-5 : Add up the four terms of step 4 :
                    (2x+1)  •  (x-1)
             Which is the desired factorization

Trying to factor as a Difference of Squares :

 2.2      Factoring:  x²⁶-6 

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 : 6 is not a square !!

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

Final result :

  (x - 1) • (2x + 1) • (x26 - 6)