You are completely on the right track except the last line
x+5/2 = sqrt(5)/2 or x+5/2 = -sqrt(5)/2
Once the middle term is not an even number I get very confused.
This is what I have so far:
x^2 + 5x + 5 = 0
x^2 + 5x + (5/2)^2 = -5 = (5/2)^2
(x + 5/2)^2 = -5 + 25/4
(x + 5/2) = -20/4 + 25/4 = 5/4
I don't know if I am on the right track or doing completely wrong.