1. **State the problem:** Solve the differential equation $$xy'' - (2 + x)y' + 4y = 0$$ with initial condition $$y(0) = 0$$ using the Laplace transform. 2. **Rewrite the equation:** The equation is not in a standard form for Laplace transform because of the variable coefficient $$x$$ multiplying $$y''$$ and $$y'$$. Laplace transform is typically used for linear ODEs with constant coefficients. Here, the presence of $$x$$ complicates direct application. 3. **Check initial conditions:** We have $$y(0) = 0$$ but no initial condition for $$y'(0)$$, which is needed for Laplace transform. 4. **Conclusion:** This equation is a Cauchy-Euler type or variable coefficient ODE, not suitable for Laplace transform directly. Instead, it is better solved by substitution or series methods. Since the user specifically requests Laplace transform, but the equation is not suitable for it, we cannot apply Laplace transform directly. **Final answer:** The given equation cannot be solved by Laplace transform due to variable coefficients and insufficient initial conditions.