The rate of change of the objects height with time is the first derivative of the function s(t) this also (intuitively) would be the vertical velocity of the balloon.
When the balloon has a zero velocity it's reached the top of its "curve" - its maximum height.
So if $s(t)=-18t^2+120t$ then $s'(t)=-36t+120$ now we must solve this for zero to find its maximum height or evaluate it at 2 or 4 for its velocity after 2 or 4 seconds (I'll leave that last one for you, it's trivial).
when the balloon "touches" back down to earth (more aptly it crashes) the height is zero
$$s(t)=0=-18t^2+120t$$ Right away this tells us that at time 0 the balloon is already at zero height. We're more interested in later on, so we divide by $t$ (knowing it's not zero anymore) then we get.
In doing physics questions it's useful to write down your knowns (quantities that are given to you) and the things you want to solve for. Looking at this find a way to rewrite the question in a more "step-by-step" way before you solve. Physics questions hone a very useful set of problem solving skills and enrich you with more than just a basic understanding of Newtonian Mechanics - they train you in a type of thinking that is ubiquitous and necessary in many fields of study.