$\text{ If }\alpha ,\beta ,\gamma \text{ are different from }1\text{ and the roots of }a{x}^3+b{x}^2+cx+d=0\text{ and }a\ne 0\text{ , }$$\left(\alpha -\beta \right)\left(\beta -\gamma \right)\left(\gamma -\alpha \right)=\frac{15}{2}\text{ and if }\left|\begin{array}{ccc}\frac{\alpha }{1-\alpha }& \frac{\beta }{1-\beta }& \frac{\gamma }{1-\gamma }\\ \alpha & \beta & \gamma \\ {\alpha }^2& {\beta }^2& {\gamma }^2\end{array}\right|=\frac{pd}{a+b+c+d}\text{ . Then greatest }$$integer less than \left|p\right| is$

$\begin{array}{l}\mathrm{\Delta }=\frac{\alpha \beta \gamma }{(1-\alpha )(1-\beta )(1-\gamma )}\left|\begin{array}{ccc}1& 1& 1\\ 1-\alpha & 1-\beta & 1-\gamma \\ \alpha (1-\alpha )& \beta (1-\beta )& \gamma (1-\gamma )\end{array}\right|\\ \\ \end{array}$

$R_1\to R_1-R_{2,}$

$=$$\frac{\alpha \beta \gamma }{(1-\alpha )(1-\beta )(1-\gamma )}\left|\begin{array}{ccc}1& 1& 1\\ \alpha & \beta & \gamma \\ \alpha {\left(1-\alpha \right)}^{ }& \beta {\left(1-\beta \right)}^{ }& \gamma {\left(1-\gamma \right)}^{ }\end{array}\right|$  

$R2\to R_2-R_3$

 

$\frac{\alpha \beta \gamma }{(1-\alpha )(1-\beta )(1-\gamma )}\left|\begin{array}{ccc}1& 1& 1\\ \alpha & \beta & \gamma \\ {\alpha }^2& {\beta }^2& {\gamma }^2\end{array}\right|$

 

$=\frac{\alpha \beta \gamma (\alpha -\beta )(\beta -\gamma )(\gamma -\alpha )}{(1-\alpha )(1-\beta )(1-\gamma )}$

$=\frac{\frac{-d}{a}\times \frac{15}{2}}{\frac{a+b+c+d}{a}}=\frac{\frac{-15}{2}d}{a+b+c+d}$

$\left|p\right|=7.50$