factor the polynomial completely,clearly show GCF 
4x² - 16 
I went another step and I am not sure weather 
to factor again after this: 
(2x - 4)(2x + 4)

What you did was mathematically correct, but 
the result is not simplified.

You skipped the first step which should always
be done first.

Before looking for any other factoring method, 
always look for a common factor.

In doing your problem

    4x² - 16

You should first have noticed that there is 
a common factor of 4.  You should have done 
that first. Then you would have

   4(x² - 4)

Then you would factor the expression in the
parentheses as the difference of two squares
and the final answer would have been:

  4(x - 2)(x + 2).

Now what you did was skip the first step and
factor the original problem as the difference
of two squares, and got

  (2x - 4)(2x + 4)

That can be factored further but it takes more
steps than if you had factored out the 4 first:

Factor out 2 in the first parentheses:

 2(x - 2)(2x + 4)

Now factor out 2 in the second parentheses:

 2(x - 2)2(x + 2)

Now multiply the 2's together and get
 
 4(x - 2)(x + 2)

Although that is correct it is going about 
it the long way.

ALWAYS look for a common factor FIRST!

[Incidentally you originally had "(2x-4)(2x-4)".
not (2x-4)(2x+4). I assumed that was a typo 
and so I corrected it. I assumed you knew that 
the sign in the second parentheses should have
been +, not -.  If you thought they should both 
be -, learn now that one sign is - and the 
other is +.]

Edwin