factoring $(x^6 - y^6)$: what is going on here? I apologize if this question already exists, but it was quite difficult to word.
In my math class today, we learned how to factor a difference of two perfect cubes. One of our practice questions was:
factor $x^6 - 64$
almost everyone other than me (including my teacher, as he was rushing) ended up with:
$(x^2 - 4)(x^4 + 4x^2 + 16)$
I pointed out two things:
1:
$(x^6 - 64)$ was also a difference of two perfect squares, and we could factor it into this:
$(x^3 + 8)(x^3 - 8)$
Which could then be factored further into:
$(x + 2)(x^2 - 2x + 4)(x - 2)(x^2 + 2x + 4)$
and 2:
the $(x^2 - 4)$ was also a difference of two perfect squares, and we could factor further: 
$(x + 2)(x - 2)(x^4 + 4x^2 + 16)$
Now for my question:

How are $(x + 2)(x^2 - 2x + 4)(x - 2)(x^2 + 2x + 4)$ and $(x + 2)(x - 2)(x^4 + 4x^2 + 16)$ equivalent?

The teacher didn't seem to know off the top of his head, and I can't figure it out after trying for a half an hour. I've heard that you can factor the sum of two perfect squares with imaginary numbers, so maybe i can do something there to help explain this?
edit: fixed instances of "- 16" to "+ 16", and "4x" to "4x^2" (thanks for pointing that out)

"How are $(x+2)(x^2−2x+4)(x−2)(x^2+2x+4)$ and $(x+2)(x−2)(x^4+4x^2+16)$ equivalent?"
Because $(x^2−2x+4)(x^2+2x+4)=x^4+4x^2+16$
===
Nother way of looking at it.
$a^3-b^3=(a-b)(a^2+ab+b^2) $
$a^2-b^2=(a-b)(a+b) $
So therefore $a^6-b^6=(a^2-b^2)(a^4+a^2b^2+b^4) =(a-b)(a+b) (a^4+a^2b^2+b^4)$
Yet, $a^6-b^6=(a^3-b^3)(a^3+b^3)=(a-b)(a^2+ab+b^2)(a^3+b^3) $
So it must be that $ (a+b) (a^4+a^2b^2+b^4)= (a^2+ab+b^2)(a^3+b^3) $ somehow.
So $(a+b)|(a^2+ab+b^2) $ or $(a+b)|(a^3+b^3) $.
$a^2+ab+b^2=a (a+b)+b^2$ which doesn't seem to work.
$a^3+b^3=a^2 (a+b)-a^2b+b^3=a^2(a+b)-ab (a+b)+ab^2+b^3=a^2 (a+b)-ab (a+b)+b^2 (a+b)-b^3+b^3=(a+b)(a^2-ab+b^2) $
So now we must have $ (a+b) (a^4+a^2b^2+b^4)= (a+b)(a^2+ab+b^2)(a^2-ab+b^2)$ and so it must be that:
$a^4+a^2b^2+b^4=(a^2+ab+b^2)(a^2-ab+b^2)$
And if we try to multiply it out, we see that it is so.
I don't blame the students for not seeing this, but the teacher ought to have recognize that it had to happen.
