The coefficient of $x^2$  in $\left(x-1\right)\left(x-5\right)\left(x-3\right)=0$  is

$\left(x^2-6x+5\right)\left(x-3\right)=0$

$x^3-3x^2-6x^2+18x+5x-15=0$

$x^3-9x^2-23x-15=0$

$\text{coefficient}\text{of}{x}^2=-9$

$\begin{array}{c}\left(or\right)\\ \left(x-1\right)\left(x-5\right)\left(x-3\right)=0\end{array}$

$\text{roots}\text{are}1,5,3$

$\text{let}\alpha =1,\beta =5,\gamma =3$

$\text{coefficient}\text{of}{x}^2$

$=-\left(\alpha +\beta +\gamma \right)$

$=-\left(1+5+3\right)$

$=-9$