Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$\left(x \left(x + 1\right) + 1\right) e^{- x} = 0$$
Solve this equationThe points of intersection with the axis X:
Numerical solution$$x_{1} = 97.6792665182734$$
$$x_{2} = 101.656718461289$$
$$x_{3} = 87.7462973895715$$
$$x_{4} = 75.286424855108$$
$$x_{5} = 60.0896238087298$$
$$x_{6} = 107.626491356935$$
$$x_{7} = 73.8777249104058$$
$$x_{8} = 99.6677304943578$$
$$x_{9} = 56.1764128472192$$
$$x_{10} = 85.7619385838247$$
$$x_{11} = 48.4126335633387$$
$$x_{12} = 50.3434895067961$$
$$x_{13} = 77.8342933423609$$
$$x_{14} = 71.9016957478322$$
$$x_{15} = 81.7959842371517$$
$$x_{16} = 115.591713770949$$
$$x_{17} = 95.6913649521561$$
$$x_{18} = 93.7040680768252$$
$$x_{19} = 111.608393846173$$
$$x_{20} = 79.8145571065428$$
$$x_{21} = 105.63612946398$$
$$x_{22} = 69.9273869463763$$
$$x_{23} = 42.6818562990598$$
$$x_{24} = 119.576290618133$$
$$x_{25} = 35.3149822710978$$
$$x_{26} = 58.1310183346547$$
$$x_{27} = 83.7784744169568$$
$$x_{28} = 37.1091027064473$$
$$x_{29} = 62.0517182834213$$
$$x_{30} = 52.2818070493597$$
$$x_{31} = 64.0168749663061$$
$$x_{32} = 65.9847344724224$$
$$x_{33} = 46.4907116808764$$
$$x_{34} = 54.2264246629927$$
$$x_{35} = 103.646195429544$$
$$x_{36} = 44.5796215483367$$
$$x_{37} = 67.9549919282347$$
$$x_{38} = 117.583854144694$$
$$x_{39} = 121.569006742071$$
$$x_{40} = 40.8007664623713$$
$$x_{41} = 109.617254342522$$
$$x_{42} = 75.8553063627584$$
$$x_{43} = 89.7314799061846$$
$$x_{44} = 113.599887264453$$
$$x_{45} = 38.9409792435025$$
$$x_{46} = 91.7174225341436$$