$\large \begin{align*}

&f_n(x)=(nx)^2 e^{-nx}\\ \\

&f_n(x)=e^{2\ln(nx)}e^{-nx} =\\ \\&e^{2\ln(n)}e^{2\ln(x)-nx} \\ \\

&\mbox{Note that }2\ln(x)-nx<0, \forall n>0, \forall x > 0\mbox{ so } \\ \\

&e^{2 \ln(x)-nx}<1\mbox{ and thus } \\ \\

&f_n(x)<e^{2\ln(n)}

\end{align*}$

You do need to prove the assertion $2\ln(x)-nx<0, \forall n, \forall x \geq 0$

I leave that to you.