Auxillary Equation with real and dinstinct root

Please solve the following equation :-

$\displaystyle 2\ddot{x} + 5\dot{x} - 12x = 0$

We have only been taught to solve the equations of $\displaystyle y'$ or $\displaystyle y''$ sort. I managed to get the numerical value but I couldn't figure out that how did $\displaystyle t$ in $\displaystyle e^{(3/2)t}$ come ? What's the general equation of this type ? And please also solve the equation step by step as I want to know the *real* way of solving these type of equations.