To decide if a function is increasing or decreasing over an interval, we have to make the first derivative test, meaning that we have to discuss the sign of f'(x).
Differentiating our function, f(x), we'll have:
f'(x) = [ln(1+x)]' - [2x/(x+2)]'
f'(x) = 1/(1+x) -[(2x)'(x+2)-(2x)(x+2)']/(x+2)^2
f'(x) = 1/(1+x) -(2x+4-2x)/(x+2)^2
In order to compute the sum of 2 ratios, we'll have to have the same denominator.
f'(x) = [(x+2)^2-4(1+x)]/(x+2)^2
f'(x) = (x^2+4x+4-4-4x)/(x+2)^2
f'(x) = x^2/(x+2)^2
Both, numerator and denominator, are positive, so f'(x)>0 over the interval [0,+inf).
If f'(x)>0. the function f(x) is increasing over the same interval [0,+inf).