Fourier Transform involving frequency differentiation property

Question

Going through some past exams, I've came across this question;

To get started, I've noticed that it resembles to the frequency differentiation property;

Now, the differentiation was extremely messy, and as a result, the second derivative was worse than the first

Is there another property that I can use to simplify this problem or the only is using this derivative? The formula sheet we used for this paper is here and here

Answer 1

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.