Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  ((4•(y4))+(5•(y3)))-(((6•(y3))-2y)+23y2)
 Step  2  :

Equation at the end of step  2  :

  ((4•(y4))+(5•(y3)))-(((2•3y3)-2y)+23y2)
 Step  3  :

Equation at the end of step  3  :

  ((4 • (y4)) +  5y3) -  (6y3 + 8y2 - 2y)
 Step  4  :

Equation at the end of step  4  :

  (22y4 +  5y3) -  (6y3 + 8y2 - 2y)

Step  5  :

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   4y⁴ - y³ - 8y² + 2y  = 

  y • (4y³ - y² - 8y + 2) 

Checking for a perfect cube :

 6.2    4y³ - y² - 8y + 2  is not a perfect cube

Trying to factor by pulling out :

 6.3      Factoring:  4y³ - y² - 8y + 2 

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1:  -y² + 2 
Group 2:  4y³ - 8y 

Pull out from each group separately :

Group 1:   (-y² + 2) • (1) = (y² - 2) • (-1)
Group 2:   (y² - 2) • (4y)
               -------------------
Add up the two groups :
               (y² - 2)  •  (4y - 1) 
Which is the desired factorization

Trying to factor as a Difference of Squares :

 6.4      Factoring:  y² - 2 

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 : 2 is not a square !!

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

Final result :

  y • (4y - 1) • (y2 - 2)