Fourier transform of \$x(t) =t^2e^{-|t|}\$
Your representation of the negative branch of the function is incorrect.Try putting  $$t=-\infty $$ to see that your function diverges as opposed to the definition.The correct integral is as stated:
$$X(\omega)=\int_{0}^\infty t^2 exp(-j\omega t-t) dt +\int_{-\infty}^0 t^2 exp(-j\omega t+t)dt$$ (In most mathematical problems you might find $$\frac{1}{(\sqrt{2\pi})}$$ multiplied to the integral,so do adjust that for yourself.)

I think you are talking about trying to evaluate it from your available results.
In that case try writing 
$$f(t)=t^2e^{-t}u(t)+t^2e^tu(-t)$$
You can use the results you have stated along with the time scaling property for the required integral.

I think this is a homework problem,so I leave the final values for you to figure out.