Đáp án:

\( x(x+1)f'(x) - f(x) = x^2(2x+3) \)

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

\( \Rightarrow \left( \frac{x+1}{x} \right)f'(x) - \frac{1}{x^2}f(x) = 2x+3 \)

\( \Rightarrow \left( \frac{x+1}{x}f(x) \right)' = 2x+3 \)

\( \Rightarrow \frac{x+1}{x}f(x) = x^2 + 3x + C \)

\( f(1) = 2 \Rightarrow C = 0 \Rightarrow f(x) = \frac{x^3 + 3x^2}{x+1} \)

\( \Rightarrow f(2) = \frac{20}{3} \Rightarrow \boxed{C}\)