Solving ODE using Laplace transformation without table of transform

Question

Solve $$y'' + 4y' -5y = 2xe^{-3x}$$ using the Laplace transform method. For the the Laplace transform of the ODE you must do the integration (do not use table).

Table can be only used when computing the inverse Laplace transform.

So, I started with that:

$$y'' + 4y' -5y = 2xe^{-3x}$$

If $y = e^{ax}$ , then $y'' = ae^{ax}$

inf int $0 = ae^{ax}\cdot e^{-sx} dx$