Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  (((8 • (w3)) +  (2•3w2)) -  4w) -  3
 Step  2  :

Equation at the end of step  2  :

  ((23w3 +  (2•3w2)) -  4w) -  3

Step  3  :

Checking for a perfect cube :

 3.1    8w³+6w²-4w-3  is not a perfect cube

Trying to factor by pulling out :

 3.2      Factoring:  8w³+6w²-4w-3 

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

Group 1:  -4w-3 
Group 2:  8w³+6w² 

Pull out from each group separately :

Group 1:   (4w+3) • (-1)
Group 2:   (4w+3) • (2w²)
               -------------------
Add up the two groups :
               (4w+3)  •  (2w²-1) 
Which is the desired factorization

Trying to factor as a Difference of Squares :

 3.3      Factoring:  2w²-1 

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 :  2  is not a square !!

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

Final result :

  (2w2 - 1) • (4w + 3)