Bounding the derivative of the auxiliary function g

Let g be the function defined on [0 ; +∞[ by

   g(x) = ln(1 + x) − 2x/(2 + x)

After justifying that g is differentiable, prove that for every x ≥ 0:

   0  ≤  g'(x)  ≤  x²/4

(Question I.2.a of the assignment.)