Answer
$$2\ln \left| {2{x^2} + 3x} \right| + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{8x + 6}}{{2{x^2} + 3x}}} dx \cr
& {\text{substitute }}u = 2{x^2} + 3x,{\text{ }}du = \left( {4x + 3} \right)dx{\text{ }} \cr
& {\text{ }}dx = \frac{{du}}{{4x + 3}} \cr
& \int {\frac{{8x + 6}}{{2{x^2} + 3x}}} dx = \int {\frac{{8x + 6}}{u}} \frac{{du}}{{4x + 3}} \cr
& = \int {\frac{{2\left( {4x + 3} \right)}}{u}} \frac{{du}}{{4x + 3}} \cr
& = 2\int {\frac{{du}}{u}} \cr
& {\text{find the antiderivative}} \cr
& = 2\ln \left| u \right| + C \cr
& {\text{ with}}\,\,\,u = 2{x^2} + 3x \cr
& = 2\ln \left| {2{x^2} + 3x} \right| + C \cr} $$