= Solution
Let $u=x^{-5}$, which tends to zero as $|x|\to\infty$. Factor the denominator and use the <cosecant expansion near zero>:
$$
g(x)=x^2\frac{u}{\sin u},\qquad
\frac{u}{\sin u}=\left(1-\frac{u^2}{6}+\frac{u^4}{120}+O(u^6)\right)^{-1}
=1+\frac{u^2}{6}+\frac{7u^4}{360}+O(u^6).
$$
Substitution gives the first two nonzero terms and the requested omitted order:
$$
\boxed{g(x)=x^2+\frac1{6x^8}+O(x^{-18}),\qquad |x|\to\infty.}
$$
More precisely the next term is $7/(360x^{18})$, followed by $O(x^{-28})$. The outside factor $x^2$ is essential when converting the remainder order from powers of $u$ to powers of $x$.
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