Lets say I have: 

$\ y''(t) + 3y'(t) + 4y(t)$ = 0

with characteristic equation

$\ x^2 + 3x + 4$ = 0

What implications about the plot of the solution to the diffeq can be drawn from the characteristic equation? I know that the coefficient to $\ y'(t)$ is the damping term and we can determine if it is over/critically damped by comparing it to a function of the coefficient to the $\ y(t)$ term (don't remember it off the top of my head). What else am I missing? 

UPDATE: I know how to solve the diffeq. My question is what implications about the solution can be made from the characteristic equation BEFORE solving all the way. I mean in terms of damping, convergence, resonance, etc.