Letting $n \to \infty$, the LHS grows like $\alpha \beta n^2+\mathcal{O}(n)$, similarly for the RHS and it easily follows that $\alpha\beta=\gamma\delta$. Now writing $\lfloor x\rfloor=x-\{x\}$, we find that \[-\frac{2}{n} \le \beta\{n\alpha\}+\alpha\{n\beta\}-\gamma\{n\delta\}-\delta\{n\gamma\} \le \frac{2}{n}.\]But for $\alpha$ irrational, the numbers $\{n\alpha\}$ are equidistributed modulo $1$. So the average value of the term in the middle is \[\frac{\beta+\alpha-\gamma-\delta}{2}.\]In particular, the inequality can only hold for all $n$ if $\alpha+\beta=\gamma+\delta$. Since we also know $\alpha\beta=\gamma\delta$, the claim now follows from Vieta.