Is there a more elegant way to prove that $\sum_{n=1}^{\infty}{\frac{n^3+2n^2+1}{n^4-10n-7}}$ diverges? 
Is there a more elegant way to prove that
  $\sum_{n=1}^{\infty}{\frac{n^3+2n^2+1}{n^4-10n-7}}$ diverges?

Here's my (to my mind) messy proof using the comparison test:$$\frac{n^3+2n^2+1}{n^4-10n-7}\geq \frac{n^3}{n^4-10n-7}=\frac{1}{n-\frac{10}{n^2}-\frac{7}{n^3}}\geq \frac{1}{n} \Longleftrightarrow n\geq n-\frac{10}{n^2}-\frac{7}{n^3} \geq 0$$
since the harmonic series diverges, the aforementioned series diverges. Is there a more elegant way? Maybe a neater diverging series to compare it to? I can't use any tests beside the quotient, root and comparsion test.
Especially these rational polynomials really bother me!

For $n>17$, $0<n^4-10n-7<n^4$ and $n^3+2n^2+1>n^3,$ so $$\frac{n^3+2n^2+1}{n^4-10n-7}>\frac{n^3}{n^4}=\frac{1}{n}$$ for $n>17.$
