Factorize 2x^4 - 24x^2 - 128

We have the function

2x^4 - 24x^2 - 128 
= 2 (x^4 - 12x^2 - 64)
For the sake of simplicity, assume x^2 = t.
= 2 ((x^2)^2 - 12(x^2) - 64)
= 2 (t^2 - 12t - 64)
Now, t^2 - 12t - 64 can be easily factorized be splitting the middle term.
= 2 (t^2 + 4t - 16t - 64)
= 2 (t(t + 4) - 16(t + 4))
= 2 (t + 4)(t - 16) 

Putting back x^2 = t

= 2 (x^2 + 4)(x^2 - 16)
= 2 (x^2 + 4)(x^2 -4^2)
= 2 (x^2 + 4)(x + 4)(x - 4)