Fourier Transform involving frequency differentiation property
I would treat it as two step functions added together by splitting the absolute value.
t^2*exp(-4*(t-5))*u(t-5) + t^2*exp(-4*-(t-5))*u(-t+5) - dirac(t-5)

Then you can use the "nth-order time-rising causal delay exponential function" transform. Since they're added together, you can use superposition. The dirac part of it removes the discontinuity that would occur at t=5.