It's a method called "completing the square". We add/subtract an appropriate constant term that allows us to condense three terms into a squared binomial, and leaves a constant remainder outside the squared term.
In this case, we first factor out -16 from the first two terms:
When we square a binomial like , we end up with the expansion
, so that in the quadratic above we need
with
and
. Then the constant term we have to add/subtract will be
:
Now the first three terms within the parentheses can be condensed to get
and all that's left to do is distribute a factor of -16 and combine the remaining constant term:
Then the vertex of the parabola defined by this expression occurs at .