Given g(x) = x^2 + 2x + 5 how would I complete the square

we have:

x² + 2x + 5

forget about the +5 for now (we'll add it back later), focus on

x² + 2x

We want this to look like (x + a)² = x² + 2ax + a²

match the coefficient for 2ax --> that is 2a should match x's coefficient (which is 2...from the 2x)

so what must a be?

a = 1
-->

(x + 1)² = x² + 2 * 1 * x + 1² = x² + 2x + 1

but that's NOT x² + 2x...how can we make them the same...simple subtract 1 from the above:

(x² + 2x + 1) - 1 = x² + 2x + 0 = x² + 2x
-->

x² + 2x = (x + 1)² - 1
--> now plug that into the original

g(x) = x² + 2x + 5 = (x + 1)² - 1 + 5 = (x + 1)² + 4

In order to complete the square, you should isolate the x terms on one side and the constant on the other, so set g(x)=0 and then rewrite x^2+2x=-5.
Then, for the actual "completing the square" part, you halve the number in front of x, which in this case is 2, and square the result. (0.5*2)^2=1.
Now, take the 1 and add it to both sides of the equation, so you have x^2+2x+1=-5+1.
After the process of completing the square, you can simplify to (x+1)^2=-4.
Finally, move the constant back on the same side of the equation, and you have g(x)=(x+1)^2-4.

Theodore Roosevelt