For example here is the problem: $(t^2 \cos{\omega t})u(t)$
I have to find it using laplace transform; here is what I think it is,
I have $t^2$$(\cos{\omega t})u(t)$ which I think I can solve them separate and them put the result together such as:
laplace transform of $t^n = n!/s^\left(n+1\right)$ so $t^2 = 2!/s^\left(2+1\right)$ right?
Now laplace transform for $(\cos\omega t)u(t) = s/s^2+\omega^2$ them
$(\cos{\omega t})u(t)= s/s^2+w^2$ so now I think I can put all these together to come up with the answer?? any help please
Hint:
You might want to use this property of the Laplace transform:
$$\large\mathcal{L}(t^nf(t))=(-1)^n\dfrac{d^n\mathcal{L}(f(t))}{ds^n}$$
Where, in your case, you have:
$$\cases{n=2 \\ f(t)=\cos(\omega t)\cdot u(t)}$$
Using this, can you come up with the result ?
Note: And by the way, regarding the method you tried to use:
$$\large\mathcal{L}\left(f(t)\cdot g(t)\right)\neq\mathcal{L}\left(f(t)\right)\cdot\mathcal{L}\left(g(t)\right)$$