If $\alpha ,\beta$  be the roots of $x^2+px+q=0$  and $\alpha +h,\beta +h$  are roots of $x^2+rx+s=0$ , then :

$\alpha +\beta =-p,\alpha \beta =q$

                  $\alpha +\beta +2h=-r$ ,

         $\left(\alpha +h\right)\left(\beta +h\right)=s$

                        $-p+2h=-r$

$\Rightarrow$                  $h=\frac{p-r}{2}$

Now,          $\alpha \beta +h\left(\alpha +\beta \right)+h^2=s$

$\Rightarrow$      $q+\left(\frac{p-r}{2}\right)\left(-p\right)+{\left(\frac{p-r}{2}\right)}^2=s$

$\Rightarrow$     $q-\frac{\left(p^2-pr\right)}{2}+\frac{p^2+r^2-2pr}{4}=s$

$\Rightarrow$     $4q-2p^2+2pr+p^2+r^2-2pr=4s$

$\Rightarrow$                 $4q-p^2+r^2-4s=0$

$\Rightarrow$                       $r^2-4s=p^2-4q$