sequences and series - Show that $sumlimits_{n=1}^inftyfrac{2n^2-1}{3n^5+2n+1}$ converges or diverges

I'm working with some infinite series problems and I have to show that the series
$\sum\limits_{n=1}^\infty\dfrac{2n^2-1}{3n^5+2n+1}$ converges or diverges. I don't have a lot of experience doing this yet, and this is a problem that my teacher made up so I have no way to check my answer.

For this problem, I said the series converges by direct comparison with $\dfrac{2}{3}\cdot\sum\limits_{n=1}^\infty\dfrac{1}{n^3}$ which converges by the p-series test. However, I'm not sure if I did this correctly and if all of the steps I took were "legal". This is my reasoning:

$\dfrac{2n^2-1}{3n^5+2n+1} \le \dfrac{2n^2}{3n^5}$ for all n.

$b_n = \dfrac{2n^2}{3n^5}=\dfrac{2}{3n^3}$

$\sum\limits_{n=1}^\infty\dfrac{2}{3n^3} =\dfrac{2}{3}\cdot\sum\limits_{n=1}^\infty\dfrac{1}{n^3}$ which converges by the p-series test.

Therefore, $\sum\limits_{n=1}^\infty\dfrac{2n^2-1}{3n^5+2n+1}$ converges by the direct comparison test with $\sum\limits_{n=1}^\infty\dfrac{1}{n^3}$.

Could anyone verify I used the test correctly or point out my mistakes? Thank you.