Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  ((x4) -  (2•13x2)) +  25

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  x⁴-26x²+25 

The first term is,  x⁴  its coefficient is  1 .
The middle term is,  -26x²  its coefficient is  -26 .
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   -26 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -25  and  -1 
                     x⁴ - 25x² - 1x² - 25

Step-4 : Add up the first 2 terms, pulling out like factors :
                    x² • (x²-25)
              Add up the last 2 terms, pulling out common factors :
                     1 • (x²-25)
Step-5 : Add up the four terms of step 4 :
                    (x²-1)  •  (x²-25)
             Which is the desired factorization

Trying to factor as a Difference of Squares :

 2.2      Factoring:  x²-1 

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 : 1 is the square of 1
Check :  x²  is the square of  x¹ 

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

Trying to factor as a Difference of Squares :

 2.3      Factoring:  x² - 25 

Check : 25 is the square of 5
Check :  x²  is the square of  x¹ 

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

Final result :

  (x + 1) • (x - 1) • (x + 5) • (x - 5)