The differential operator $(D + 6)^3$ is supposed to annihilate the function $x^{2}e^{-6x}$, we will check that:
$(D)(x^{2}e^{-6x}) =$
$(D + 6)(x^{2}e^{-6x}) =$
$D((D + 6)(x^{2}e^{-6x})) =$
$(D + 6)^2(x^{2}e^{-6x}) =$
$D((D + 6)^2(x^{2}e^{-6x})) =$
$(D + 6)^3(x^{2}e^{-6x}) =$