\(I= \int _{-1}^{1}\frac{x^{2} -3\left|x\right|+2}{x+2} dx \)
\(\int _{-1}^{1}\frac{x^{2} -3\left|x\right|+2}{x+2} dx =\int _{-1}^{0}\frac{x^{2} +3x+2}{x+2} dx +\int _{0}^{1}\frac{x^{2} -3x+2}{x+2} dx \)
\(=\int _{-1}^{0}\left(x+1\right)dx +\int _{0}^{1}\left(x-5+\frac{12}{x+2} \right)dx \)
\(=\frac{1}{2} +\left(\frac{1}{2} x^{2} -5x+12\ln \left|x+2\right|\right)\left|\begin{array}{l} {1} \\ {0} \end{array}\right. \)
\(
=\frac{1}{2} +\left(\frac{1}{2} -5+12\ln 3\right)-12\ln 2=12\ln \frac{3}{2} -4 \)