Completing the square is a technique for converting a quadratic polynomial of the form $\displaystyle ax^2+bx+c$ into the form $\displaystyle a(x-p)^2+q$ where $\displaystyle p,q$ are constants.
So that's what we've done here and notice $\displaystyle (x-4)^2-16=x^2-8x+16-16=x^2-8x$.
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