Q:

How to determine number of roots and what type for quartic equations?

I want to figure out the number of roots and their types in the quartic equation $$x^4-34x^2-x+272=0$$ without actually solving it. Is there such a way to do so?

A:

With this particular equation, you can let $f(x) = x^4 - 34x^2 - x + 272$ (this is just for convenience), and then calculate:
\begin{align*}
f(-5) &= 52 \\
f(-4) &= -12 \\
f(0) &= 27 \\
f(4) &= -20 \\
f(5) &= 42
\end{align*}
Since this function should look "continuous" when you draw it, and it changes sign between each of -5, -4, 0, 4, 5, this implies that there is a root in between any adjacent pair of each of these numbers, so we have at least two negative real roots and two positive real roots. I suspect that's what your sign changing business is supposed to mean.

