[Kevin Sevcik]

(1) comes from dividing ax^2 + bx + c by a:
ax^2 + bx + c = a*(ax^2/a + b/a*x + c/a)

a perfect square is a perfect square factor, I just picked a name. (x - h)^2 is a square, as opposed to (x - h) * (x + q) which is not a square. you're looking for a term like (x - h)^2, so you expand that out to what it equals, x^2 - 2hx + h^2

at that point you say, "hey, -2hx and b/a*x are both multiplied by x. I'll make them equal"

Edit: Mike, I knew there was a better name, but I've long since forgotten.