Quadratic sum..

\hspace{-16}$Here $\mathbf{x=\alpha,\beta}$ be the roots of the equation $\mathbf{x^2+px+q=0}$\\\\ So $\mathbf{(x-\alpha)}$ and $\mathbf{(x-\beta)}$ be the factor of $\mathbf{x^2+px+q}$\\\\ So $\mathbf{x^2+px+q=(x-\alpha).(x-\beta)}$\\\\ So Put $\mathbf{x=\gamma},$ We Get\\\\ $\mathbf{\gamma^2+p\gamma+q=(\gamma-\alpha).(\gamma-\beta)=(\alpha-\gamma).(\beta-\gamma)........................(1)}$\\\\ Similarly $\mathbf{x=\delta}\;,$ We Get\\\\ $\mathbf{\delta^2+p\delta+q=(\delta-\alpha).(\delta-\beta)=(\alpha-\delta).(\beta-\delta)..........................(2)}$\\\\ So $\mathbf{(\alpha-\gamma).(\beta-\gamma).(\alpha-\delta).(\beta-\delta)=\left(\gamma^2+p\gamma+q\right).(\delta^2+p\delta+q)}$\\\\ $\mathbf{=(\gamma.\delta)^2+P.\gamma. \delta.(\delta+\gamma)+q\left\{(\gamma+\delta)^2-2\gamma.\delta\right\}+pq.(\gamma+\delta)+q^2}$\\\\ $\mathbf{=s^2-P.r.s+q.(r^2-2s)-pqr+q^2}$\\\\

what is the ans of 2nd part???

\hspace{-16}$If Given equation has Only one Real Root say $\mathbf{x=\alpha}$ Then\\\\ $\mathbf{\alpha^2+p\alpha+q=0...........................(1)}$\\\\ $\mathbf{\alpha^2+r\alpha+s=0............................(2)}$\\\\ So $\mathbf{\alpha.(p-r)=s-p\Leftrightarrow \alpha =\frac{s-p}{p-r}}$\\\\ Now Put value of $\mathbf{\alpha}$ in equation..$\mathbf{(1)}$\\\\ $\mathbf{\left(\frac{s-p}{p-r}\right)^2+p.\left(\frac{s-p}{p-r}\right)+q=0}$