Solution - Factoring multivariable polynomials

Step  1  :

Equation at the end of step  1  :

  ((2•(m4))+((4•(m2))•(n2)))-24n4
 Step  2  :

Equation at the end of step  2  :

  ((2 • (m4)) +  (22m2 • n2)) -  24n4
 Step  3  :

Equation at the end of step  3  :

  (2m4 +  22m2n2) -  24n4

Step  4  :

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   2m⁴ + 4m²n² - 16n⁴  =   2 • (m⁴ + 2m²n² - 8n⁴) 

Trying to factor a multi variable polynomial :

 5.2    Factoring    m⁴ + 2m²n² - 8n⁴ 

Try to factor this multi-variable trinomial using trial and error 

 Found a factorization  :  (m² + 4n²)•(m² - 2n²)

Trying to factor as a Difference of Squares :

 5.3      Factoring:  m²-2n² 

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 :

  2 • (m2 + 4n2) • (m2 - 2n2)