First shifting property (Laplace transform)

Question

$$\mathcal{L}(e^{3t}\sin(5t)\sin(3t)) $$

I tried it. But I dont know the answer for this question. So I need to verify it.

Answer 1

$$\mathcal{L}(e^{3t}\sin(5t)\sin(3t))$$ $$f(t)=e^{3t}\sin(5t)\sin(3t)$$ Transform the product into a sum $$\sin(5t)\sin(3t)=\frac12 ( \cos (2t)-\cos (8t))$$ So that we have: $$f(t)=\frac 12e^{3t} \cos (2t)-\frac 12 e^{3t}\cos(8t)$$ $$\mathcal{L}(f(t))=\frac 12\mathcal{L}(e^{3t} \cos (2t))-\frac 12 \mathcal{L}(e^{3t}\cos(8t))$$ Apply the following formula: $$\mathcal{L}(e^{bt} \cos (at))=\dfrac {(s-b)}{(s-b)^2+a^2}$$