Chọn B
Ta có \(A\left(-2;0\right),O\left(0;0\right),B\left(3;2\right)\) thuộc đồ thị hàm số \(y=g\left(x\right).\)
\(\Rightarrow \left\{\begin{array}{l} {4a-2b+c=0} \\ {c\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, =0} \\ {9a+3b+c=2} \end{array}\right. \Rightarrow \left\{\begin{array}{l} {a=\frac{2}{15} } \\ {b=\frac{4}{15} } \\ {c=0} \end{array}\right. \Rightarrow \left(P\right):y=g\left(x\right)=\frac{2}{15} x^{2} +\frac{4}{15} x.\)
Suy ra:
\({I=\int _{-5}^{3}f\left(x\right)dx =\int _{-5}^{3}\left[f\left(x\right)-g\left(x\right)\right]dx +\int _{-5}^{3}g\left(x\right)dx } \)
\(=\int _{-5}^{-2}\left[f\left(x\right)-g\left(x\right)\right]dx\) \(+\int _{-2}^{0}\left[f\left(x\right)-g\left(x\right)\right]dx \)\(+\int _{0}^{3}\left[f\left(x\right)-g\left(x\right)\right]dx \)
\(+\int _{-5}^{3}\left(\frac{2}{15} x^{2} +\frac{4}{15} x\right)dx \) \(=S_{1} -S_{2} +S_{3} +\frac{208}{45} =m-n+p+\frac{208}{45} .\)