What Did I Miss in My Derivative Calculation Using First Principles?

[-Dragoon-]

Homework Statement

Find the derivative of f(x) = x^2+2x+1

Homework Equations

f(x + h) - f(x) / h
lim(h->0) f (x+h) - f(x) / h

The Attempt at a Solution

Hi everyone. I keep calculating the derivative for this function incorrectly. I haven't learned the rules of derivatives yet, I am only using first principle definition. So here's my mathematical attempt:
f (x+h) - f (x) / h = (x+h)^2 + 2(x+h)+1 - (x^2+2x+1) / h
First I expand the brackets
=> x^2+2xh+h^2+2x+1-(x^2+2x+1) / h
Now I open the brackets for f(x):
=> x^2+2xh+h^2+2x+1-x^2-2x-1) / h
Now I cancel some variables out and have left:
=> h^2+2xh / h
Now I factor out h and get rid of the fraction:
=> h(h + 2x) / h
Now I use the limit equation lim(h->0) f (x+h) - f(x) / h:
=> lim(h->0) h+2x = 2x is the derivative I calculated for. But checking my answers through online derivative calculators say the derivative is 2x+2? What have I done wrong in my calculations? I've checked them over more times than I can remember to count.

 

[fzero]

f (x+h) - f (x) / h = (x+h)^2 + 2(x+h)+1 - (x^2+2x+1) / h
First I expand the brackets
=> x^2+2xh+h^2+2x+1-(x^2+2x+1) / h

You missed a 2h in the numerator coming from the bold term.