​ Factor the polynomial function over the complex numbers.

I'm not sure what you're taught.....but....I see that 2 is a root....so....we can use synthetic division to find the other roots

2  [    1      -1     0     -2      -4  ]

                   2    2      4       4

     ___________________

         1      1      2      2      0

Our remaining polynomial is  

x^3  + x^2  + 2x  +  2

And -1 is a root...so...performing synthetic division  again, we have

-1  [  1     1        2      2   ]

              -1        0     -2

      _________________

        1     0        2      0

The remaining polynomial is  

x^2  +  2.....  setting this to 0 to find the other roots, we have

x^2  +  2  = 0

x^2  = -2       take both roots

x = ±  √2  i

So.....the factorization is :

f(x)   =  (x - 2) (x + 1) (x -   √2 i )  ( x +   √2 i  )