Formula for completing the square? My math teacher said that this was the formula for completing the square.
Original function: $$ax^2 + bx + c$$
Completed square: $$a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a} + c$$
However, using this formula I'm not getting the same answers that I would get just by determining the stuff myself.  Is this correct?

I usually find that completing the square by hand for each example is better than just using the formula.
For your example,
$$4x^2 + 4x + 5 = 4\left(x^2 + x + \frac{5}{4}\right).$$
This is of the form $x^2 + x + \mathrm{const}$, so you to find a number $a$ such that $(x+a)^2 = x^2 + x + \mathrm{const}$.  The solution is $a=\frac{1}{2}$, giving
$$4x^2 + 4x + 5 = 4\left(x+\frac{1}{2}\right)^2 + \mathrm{const}$$
Expanding the RHS, you find that the constant is $4$, so the whole expression is
$$4x^2 + 4x + 5 = 4\left(x + \frac{1}{2} \right)^2 + 4 = (2x + 1)^2 + 4$$
