The usual idea is to isolate a term on one side of the equation and find conditions under which one side dominates the other. For example,
$$ a x^q = b x^p - c$$ If $x$ is large, the left side is going to be larger than the right. Thus for $x > 0$, $b x^p - c < b x^p \le a x^q$ if $x \ge (b/a)^{1/(q-p)}$ so an upper bound on positive roots is $(b/a)^{1/(q-p)}$.
Similarly, a lower bound on positive roots is $(c/b)^{1/p}$.