If $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }$ are the roots of the equation ${\mathrm{x}}^3-{\mathrm{px}}^2+\mathrm{qx}+\mathrm{r}=0$ then $(\mathrm{\alpha }+\mathrm{\beta })(\mathrm{\beta }+\mathrm{\gamma })(\mathrm{\gamma }+\mathrm{\alpha })=$

$\begin{array}{l}\alpha ,\beta ,\gamma arerootsof{x}^3-p{x}^2+qx+r=0\\ \therefore \alpha +\beta +\gamma =p,\alpha \beta +\beta \gamma +\gamma \alpha =q,\alpha \beta \gamma =-r\\ \therefore (\alpha +\beta )(\beta +\gamma )(\gamma +\alpha )=(p-\gamma )(p-\alpha )(p-\beta )\\ =p^3-(\alpha +\beta +\gamma )p^2+(\alpha \beta +\beta \gamma +\gamma \alpha )p-\alpha \beta \gamma \\ =p^3-p^3+pq+r\\ =pq+r\end{array}$