Multiplication in a LaPlace Transform

[TG3]

Homework Statement
Find the Laplace transform of the given function, a and b are real constants.
Hint: cos(bt)=(e^ibt+e^-ibt)/2 and sin(bt)=(e^it-e^ibt)/2i.

f(t)=e^(at)*sin(bt)

The attempt at a solution

The LaPlace transform of e^at is 1/(s-a).

The LaPlace transform of sin(bt) is b/(s^2+b^2).

Simply multiply those together, I got b/s^3-as^2+bs^2-ab^2.

This is wrong, and it feel like I'm making a very basic mistake that should be obvious, doing something other than multiplying the two separate LaPlace transforms together. So, what am I supposed to do when the function I am supposed to work with has two easily distinguishable functions within it?

 

[LCKurtz]

Have you had the shifting theorems? Like for L(e^atf(t)) in terms of L(f(t))?

 

[TG3]

I don't think so, but I'm not positive... I have trouble focusing for the entire period.

 

[LCKurtz]

Well, if you haven't had the shifting theorem, you can always use the given hint. Express the sine in terms of exponentials, take the transform, and simplify it.

 

[vela]

Start by plugging that function into the definition of the Laplace transform and combining the exponentials. You might be able to see the answer immediately.