I'm lost on the first line of the explanation. If x=(x+2)x, then how does it get ([x+2]+2)[...to start? I just plugged in "(x+2)x" for x and of course got the wrong answer so am a little confused.

Justin and Russell, let's see if we can analyze this one step at a time. This type of problem is an operator problem.

<> is the operator, which causes the term inside the <..> signs, e.g., x, to transform into (x+2)x; we can verbalize the operator as: "add 2 to the thing inside the <..> sign, then multiply the result by the "thing".

So, applying <..> to x+2, can be verbalized as "add 2 to x+2, then multiply x+2 +2 by x+2"
Expressing this mathematically:
= [(x+2) +2] (x+2) = (x+4)(x+2) = x^2 + 6x +8

Similarly, for the second term:
is expressed verbally as: add 2 to (x-2), then multiply the result by (x-2); in this case the "thing" is (x-2); expressed mathematically:
= [(x-2) +2] (x-2) = x(x-2) = x^2 - 2x

Subtracting 2nd term from 1st term, we get x^2 +6x +8 - (x^2 -2x) = 8x +8 = ...

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