$I=\int {2x^2e^{x^3+2}+2xe^{2x}} \, dx\\ =2\int {x^2e^{x^3+2}} \, dx+\int {2xe^{2x}} \, dx\\ =\frac{2}{3}\int {e^{x^3+2}} \, d(x^3+2)+I_1\\ =\frac{2}{3}e^{x^3+2}+I_1+C\\ I_1=\int {2xe^{2x}} \, dx\\ u=2x=>du=2dx\\ dv=e^{2x}=>v=\frac{1}{2}e^{2x}\\ I_1=xe^{2x}-\int {e^{2x}} \, dx\\ =xe^x-\frac{1}{2}e^x+C\\ =>I=\frac{2}{3}e^{x^3+2}+xe^x-\frac{1}{2}e^x+C\\ =>m+n+p=\frac{13}{6}$
