Đáp án:

\( f(x) + x(x+1)f'(x) = (x+1)^2 \)

\(\Leftrightarrow \frac{f(x)}{x+1} + x f'(x) = x+1\)

\( \left( \frac{x}{x+1} \right)' = \frac{1}{(x+1)^2} \)

\(\Leftrightarrow \frac{1}{(x+1)^2} \cdot f(x) + \frac{x}{x+1} \cdot f'(x) = 1 \)

\(\Leftrightarrow \left( \frac{x}{x+1} \cdot f(x) \right)' =1\)

\( \frac{x}{x+1} \cdot f(x) = x + C \)

\( f(1) = 2 \Rightarrow C = 0 \Rightarrow f(x) = x+1 \)

\(\Rightarrow f(2) = 3 \Rightarrow \boxed{A}\)