Which is appropriate to use when is simply a matter of one's computational speed.
Factoring, for instance, is something we often think about only being doable when we have rational or integer coefficients, and generally smaller ones. For the most part, that's true - if you don't want to spend a painstaking amount of work, you want to focus on those cases. For instance, consider your equation:
$$h(t) = -16t^2 + 164t - 92 = 0$$
To properly factor this, you need to figure out which factors of $92\cdot -16=-1472$ have a sum of $164$. You could sit there all day enumerating the various factors and checking the sum, or you can just go "screw this" and decide to just plug-and-chug using the quadratic formula, and probably solve it faster. Really, try it - see which one you solve faster!
(Anecdote: when you see a "round to the nearest hundredth" tenet in a question, a lot of the time that means the question will have not-so-nice roots, i.e. they'll be irrational probably. Which touches on another issue: if the square root in the quadratic formula is not a perfect square, then you definitely cannot factor it just by using integers, you might have to reach for square root factors instead ... which makes this even more of a mess and is why factoring isn't always optimal. Let's nail that point home...)
Okay, so forget the big numbers and such - let's say you're great at factorizing even large numbers. But then there's a problem: factoring doesn't necessitate integers/rationals. They are easier to work with like that, but consider:
$$(x+4\sqrt 2)(x + 2\sqrt 3) = x^2 + (4 \sqrt 2 + 2 \sqrt 3) x + 8 \sqrt 6$$
At a glance, how would you even factor that latter expression, without knowing what led up to it? Or perhaps another, nastier, crueler one:
$$(x + 4 \sqrt 6)(x - 3 \sqrt 6) = x^2 + \sqrt 6 x - 72$$
At least if they're "nice enough" you can guess perhaps a means to factorize it, but I see no obvious way for the above polynomial. How would you even begin to approach this a priori?
I'd rather sit there and just plug-and-chug, than try to guess how to factor some weird expression. Simpler to do and process.
That's not to say either method is "invalid" in either situation. Factoring takes advantage of the fact that, if a polynomial of degree $n$ has roots $r_1,r_2,...,r_n$, then it can be written in the form
where $a$ is its leading coefficient. And it so happens that for integers and similar-enough polynomials, factoring can be relatively easy. But that's simply not always the case, which is why it's nice that the quadratic formula provides a quick, general solution just in case the expressions aren't nice enough - which is something we can't say for all polynomials!
In short - use whichever method you choose, both are valid. But always be cognizant of the time it takes for you to get the solution any one way, as opposed to another way. No sense in wasting time and energy when the end result is the same.