x² + ___ + 64

We know it must end up as the square of the sum of the square roots of
the given first and third terms.  The square root of x² is x and the 
square root of 64 is 8. We multiply out the square of their sum, which
is (x+8)²:

 (x+8)² = (x+8)(x+8) = x² + 8x + 8x + 64 = x² + 16x + 64.

So we see now that the middle term has to be the same as that middle term,
which is 16x:

x² + 16x + 64

You can do that or you can memorize the rule.  The middle term is always

TWICE THE PRODUCT OF THE TWO SQUARE ROOTS.

The two square roots are x and 8.  Their product is 8x and twice that is 16x.

The first way is when you can't remember the rule.

Edwin