Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$- \left(x + 8\right)^{2} e^{- x - 3} + \left(2 x + 16\right) e^{- x - 3} = 0$$
Solve this equationThe points of intersection with the axis X:
Analytical solution$$x_{1} = -8$$
$$x_{2} = -6$$
Numerical solution$$x_{1} = 71.8295111187539$$
$$x_{2} = 99.6337714382312$$
$$x_{3} = 63.9211906076499$$
$$x_{4} = 40.4724450240127$$
$$x_{5} = 50.165604409395$$
$$x_{6} = 91.6764500177863$$
$$x_{7} = 75.7917334746358$$
$$x_{8} = 69.8502200597028$$
$$x_{9} = -6$$
$$x_{10} = 105.606309881405$$
$$x_{11} = 95.6541464837633$$
$$x_{12} = 109.589786759242$$
$$x_{13} = 44.3289362867604$$
$$x_{14} = 89.6884128679652$$
$$x_{15} = 52.1211385583235$$
$$x_{16} = -8$$
$$x_{17} = 97.6437339179749$$
$$x_{18} = 34.7734798936756$$
$$x_{19} = 107.597884083738$$
$$x_{20} = 117.560325232369$$
$$x_{21} = 61.9483432722063$$
$$x_{22} = 77.774456655146$$
$$x_{23} = 59.9775817816825$$
$$x_{24} = 42.3963836086947$$
$$x_{25} = 46.2686954792205$$
$$x_{26} = 48.2145487691854$$
$$x_{27} = 65.8959068817062$$
$$x_{28} = 54.0805571210015$$
$$x_{29} = 115.567283846416$$
$$x_{30} = 32.9091112925386$$
$$x_{31} = 36.6582150315442$$
$$x_{32} = 38.558930992648$$
$$x_{33} = 58.0091582728753$$
$$x_{34} = 67.8723044353069$$
$$x_{35} = 85.7141680867988$$
$$x_{36} = 103.615084597457$$
$$x_{37} = 93.6650404271081$$
$$x_{38} = 119.553613684418$$
$$x_{39} = 56.0433676432483$$
$$x_{40} = 87.7009703512766$$
$$x_{41} = 81.7426914756113$$
$$x_{42} = 73.8100523902503$$
$$x_{43} = 101.624230413862$$
$$x_{44} = 83.7280565000326$$
$$x_{45} = 121.547136254232$$
$$x_{46} = 113.574503449427$$
$$x_{47} = 111.581999032425$$
$$x_{48} = 79.7581351198125$$