Chọn phương án B.

\(\begin{aligned}I&=\displaystyle\int\limits_1^2 \dfrac{x^2+2x}{x+1}\mathrm{\,d}x\\
&=\displaystyle\int\limits_1^2 \left(x+1-\dfrac{1}{x+1}\right)\mathrm{\,d}x\\
&=\left(\dfrac{x^2}{2}+x-\ln|x+1|\right)\bigg|_1^2\\
&=\dfrac{5}{2}+\ln2-\ln3.\end{aligned}\)

Theo đó \(a=2,\,b=2,\,c=3\).
Suy ra \(S=a-b+c=2-2+3=3\).