A Leading Number

An expression has no sides to divide, so when a is not 1 you take a out of the two x terms instead. Take 3x^2 + 6x + 4. Taking 3 out of 3x^2 + 6x gives 3(x^2 + 2x) + 4. The number 4 stays outside, untouched.

Now complete the square inside the bracket. The missing number for x^2 + 2x is 1, so x^2 + 2x is (x + 1)^2 - 1, and the expression is 3((x + 1)^2 - 1) + 4. Expanding the outer 3 gives 3(x + 1)^2 - 3 + 4, which is 3(x + 1)^2 + 1.

Watch the expanding step: the 3 multiplies the borrowed -1 as well, so the amount taken back is 3, not 1. That is the most common slip here. The vertex is (-1, 1), and a = 3 says the curve opens up, steeply.