Proving Little-Oh notation by definition
Either you do what David suggested, or you do something like this:

Let $c &gt; 0$. Then

\begin{align}
5n^2 +3n &amp;\le c(n^3-4n) &amp;\iff \\
5n+3 &amp;\le cn^2-4c &amp;\iff \\
0 &amp;\le cn^2-5n-(3+4c)
\end{align}

Now solve for $n$ and let $n_0$ be the bigger root of that term.