Help with simplifying the difference quotient

I feel like an absolute moron for asking this, but I can't seem to get my answer to match the one in my textbook. Here is the problem:

Find f(a + h), f(a), and the difference quotient (f(a + h) - f(a)) / h, where h ≠ 0. In the problem, f(x) = x / (x + 1).

I would type out my solution but I know it's wrong. I feel like such an idiot. Please help :) Thanks.

f(x) = x / (x + 1)
f(a) = a / (a + 1)

f(a + h) =>
(a + h) / (a + h + 1)


f(a + h) - f(a) =>
(a + h) / (a + h + 1) - a / (a + 1) =>
((a + h) * (a + 1) - a * (a + h + 1)) / ((a + 1) * (a + h + 1)) =>
(a^2 + a + ah + h - a^2 - ah - a) / ((a + 1) * (a + h + 1)) =>
h / ((a + 1) * (a + h + 1))

Divide by h

1 / ((a + 1) * (a + h + 1))

Let h go to 0

1 / ((a + 1) * (a + 1)) =>
1 / (a + 1)^2

f(x) = x / (x + 1)
=> f(a) = a / (a + 1)
f(a + h) = (a + h) / {(a + 1) + h}
f(a + h) - f(a) = [(a + h) / {(a + 1) + h}] - {a / (a + 1)}
= {a(a + 1) + h(a + 1) - a(a + 1) - ha)} / {(a + 1) + h}(a + 1)}
= - ha / {(a + 1)² + h(a + 1)}
=> {f(a + h) - f(a)} / h = - a / {(a + 1)² + h(a + 1)}

f(a) = a / (a+1)
f(a+h) = (a+h) / (a+h+1)
f(a+h) - f(a)
= [a/(a+1)] - [(a+h)/(a+h+1)]
= [(a² + ah + a) - (a² + ah + a + h)]/(a+1)(a+1+h)
= h / (a+h)(a+1+h)
[f(a+h)-f(a)]/ h = [h/(a+1)(a+1+h)] / h = 1 / (a+1)(a+1+h)