So for a cubic equation $ax^3 +bx^2+cx+d$ with roots $\alpha, \beta$ and $\gamma$, how do we know that
$(\alpha+\beta)(\beta+\gamma)(\alpha+\gamma)=\sum\alpha\sum\alpha\beta-\alpha\beta\gamma$
where $\sum\alpha=\alpha+\beta+\gamma$ and $\sum\alpha\beta=\alpha\beta+\beta\gamma+\alpha\gamma$
without having to actually expand $(\alpha+\beta)(\beta+\gamma)(\alpha+\gamma)$?? And even after we do expand it, how is it that we are able to see that it equals the right hand side?
Let $S=\alpha + \beta + \gamma$.
We have $a(x-\alpha)(x-\beta)(x-\gamma) = ax^3+bx^2+cx+d$.
Then $$(S-\alpha)(S-\beta)(S-\gamma) = S^3 + \frac{b}{a}S^2 + \frac{c}{a}S + \frac{d}{a}$$
But we have $\frac{b}{a} = -S$, $\frac{c}{a} = \sum\alpha\beta$, $\frac{d}{a}=-\alpha\beta\gamma$.
SO you get the conclusion.