(a) \[2x^3-x^2+6x+3=(x^2+3)(2x-1)+6.\] The quotient is \(\boxed{2x-1}\), with remainder \(\boxed6\).

(b) \[\begin{aligned}I&=\int_1^3\left(2x-1+\frac6{x^2+3}\right)dx\\&=\left[x^2-x+2\sqrt3\tan^{-1}(x/\sqrt3)\right]_1^3\\&=6+2\sqrt3(\pi/3-\pi/6)=\boxed{6+\pi/\sqrt3}.\end{aligned}\]