We will use `Lap{t^ng(t)}=(-1)^n(d^nG(s))/(ds^n)`, with `n=2`.

Now

`G(s)= Lap{sin 5t}`

`=5/(s^2+5^2)`

`=5/(s^2+25)`

The first derivative:

`d/(ds)5/(s^2+25)=(-10s)/((s^2+25)^2`

Now for the second derivative:

`d/(ds)(-10s)/((s^2+25)^2)=10(3s^2-25)/((s^2+25)^3)`

For the formula, we need:

`(-1)^2=1`

So

`Lap{t^2\ sin\ 5t}=10(3s^2-25)/((s^2+25)^3)`