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The homogeneous roots are obtained by solving the associated homogeneous equation $\ddot{x}+3\dot{x}+2x=0$. By substituting the exponential trial solution $x=e^{rt}$, the differential equation converts into the characteristic polynomial $r^2+3r+2=0$, which factors to $(r+2)(r+1)=0$, yielding the roots $r=-2$ and $r=-1$.

Conditions: The ODE has constant coefficients.; The trial solution is of the form $x=e^{rt}$.; The characteristic equation is a quadratic polynomial.

The homogeneous roots are obtained by solving the associated homogeneous equation $\ddot{x}+3\dot{x}+2x=0$. By substituting the exponential trial solution $x=e^{rt}$, the differential equation converts into the characteristic polynomial $r^2+3r+2=0$, which factors to $(r+2)(r+1)=0$, yielding the roots $r=-2$ and $r=-1$.

Conditions: The ODE has constant coefficients.; The trial solution is of the form $x=e^{rt}$.; The characteristic equation is a quadratic polynomial.