If $\mathrm{\alpha },\mathrm{\beta }$ are the roots of x² + ax – b = 0 and $\mathrm{\gamma },\mathrm{\delta }$ are the roots of x² + ax + b = 0 then $(\mathrm{\alpha }-\mathrm{\gamma })(\mathrm{\alpha }-\mathrm{\delta })(\mathrm{\beta }-\mathrm{\delta })(\mathrm{\beta }-\mathrm{\gamma })=$

Given that $\alpha .\beta$ are the roots of the equation $x^2+ax-b=0$

Hence, ${\alpha }^2+a\alpha =b.{\beta }^2+a\beta =b$

Given that $\gamma ,\delta$ are the roots of the equation $x^2+ax+b=0$

Hence, Sum of the roots is $\gamma +\delta =-a,$ and product of the roots is $\gamma \delta =b$

Consider the given expression

$\begin{array}{rcl}\left(\alpha -\gamma \right)\left(\alpha -\delta \right)\left(\beta -\gamma \right)\left(\beta -\delta \right)& =& \left[{\alpha }^2-\left(\gamma +\delta \right)\alpha +\gamma \delta \right]\left[{\beta }^2-\left(\gamma +\delta \right)\beta +\gamma \delta \right]\\ & =& \left[{\alpha }^2+a\alpha +b\right]\left[{\beta }^2+a\beta +b\right]\\ & =& \left[b+b\right]\left[b+b\right]\\ & =& \boxed{4b^2}\end{array}$