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$$x_{1} = -105.333005739691$$
$$x_{2} = -57.8505831390582$$
$$x_{3} = -111.304489897731$$
$$x_{4} = -101.354206694808$$
$$x_{5} = -121.264081496849$$
$$x_{6} = -71.6086441089967$$
$$x_{7} = -4135.94355451445$$
$$x_{8} = -89.4314266145357$$
$$x_{9} = -79.5176749819744$$
$$x_{10} = -81.4983526999876$$
$$x_{11} = -40.5801277728577$$
$$x_{12} = -48.1529659582005$$
$$x_{13} = -113.295754144651$$
$$x_{14} = -95.3899462845119$$
$$x_{15} = -36.9291459649712$$
$$x_{16} = -531.837727606964$$
$$x_{17} = -109.313594149021$$
$$x_{18} = -61.7664973306176$$
$$x_{19} = -63.72969387164$$
$$x_{20} = -77.5382332700649$$
$$x_{21} = -52.0117442489976$$
$$x_{22} = -69.6355644697641$$
$$x_{23} = -46.2378882480841$$
$$x_{24} = -97.3774570975963$$
$$x_{25} = -50.0781629625302$$
$$x_{26} = -42.4478982713749$$
$$x_{27} = -35.1676235533282$$
$$x_{28} = -65.6958216263569$$
$$x_{29} = -53.9523555154393$$
$$x_{30} = -91.4168785530278$$
$$x_{31} = -44.3351933877829$$
$$x_{32} = -117.279301972525$$
$$x_{33} = -119.271546597134$$
$$x_{34} = -85.4629939239387$$
$$x_{35} = -67.6645413261944$$
$$x_{36} = -75.5601507367764$$
$$x_{37} = -107.323090768864$$
$$x_{38} = -93.4030701679791$$
$$x_{39} = -83.4801577602678$$
$$x_{40} = -73.5835675835769$$
$$x_{41} = -51.1656821821462$$
$$x_{42} = -103.343367395508$$
$$x_{43} = -99.3655575325085$$
$$x_{44} = -55.8989228973151$$
$$x_{45} = -38.7376998275312$$
$$x_{46} = -87.4467756438286$$
$$x_{47} = -59.8066338460799$$
$$x_{48} = -115.287364920786$$