What would you like to understand?

Substitute the trial solution $x=Ate^{-t}$ and its derivatives into the ODE, factor out the common term $e^{-t}$, and collect like terms by powers of $t$. Equating the remaining constant term to the right-hand side yields $A=2$.

Conditions: The trial solution is $x=Ate^{-t}$.; The ODE is $\ddot{x}+3\dot{x}+2x=2e^{-t}$.; $e^{-t}$ is never zero, allowing it to be canceled from both sides.

Substitute the trial solution $x=Ate^{-t}$ and its derivatives into the ODE, factor out the common term $e^{-t}$, and collect like terms by powers of $t$. Equating the remaining constant term to the right-hand side yields $A=2$.

Conditions: The trial solution is $x=Ate^{-t}$.; The ODE is $\ddot{x}+3\dot{x}+2x=2e^{-t}$.; $e^{-t}$ is never zero, allowing it to be canceled from both sides.