$\alpha ,\beta$ are the roots of the equation $x^2-3x+5=0$  and $\gamma ,\delta$ are the roots of the equation $x^2+5x-3=0$ then the equation whose roots are $\alpha \gamma +\beta \delta$ and $\alpha \delta +\beta \gamma$ is

$\alpha +\beta =3$, $\alpha \beta =5$,  $\gamma +\delta =-5$,  $\gamma \delta =-3$

Sum of the roots  $=(\alpha \gamma +\beta \delta )+(\alpha \delta +\beta \gamma )$

$=(\alpha +\beta )(\gamma +\delta )=3\times (-5)=-15$

Product of the root $=(\alpha \gamma +\beta \delta )(\alpha \delta +\beta \gamma )$

$={\alpha }^2\gamma \delta +\alpha \beta {\gamma }^2+\beta \alpha {\delta }^2+{\beta }^2\gamma \delta$

$=\gamma \delta ({\alpha }^2+{\beta }^2)+\alpha \beta ({\gamma }^2+{\delta }^2)$

$=-3[{(\alpha +\beta )}^2-2\alpha \beta ]+5[{(\gamma +\delta )}^2-2\gamma \delta ]$

$=-3[9-10]+5[25+6]=158$

Required equation is  $x^2+15x+158=0$