\displaystyle\int\frac{3x-2}{x^2+4x+12}dx = \frac{3}{2}\int\frac{2x-\frac{4}{3}}{x^2+4x+12}dx
=\frac{3}{2}\int\frac{(2x+4) -4-\frac{4}{3}}{x^2+4x+12}dx