jmc

Let $a=\sqrt{3},$ $b=\sqrt{5},$ $c=\sqrt{6},$ $d=\sqrt{7},$ and $s=a+b+c+d.$ Then the given expression is $$\begin{array}{rl}& (s-2a{)}^2+(s-2b{)}^2+(s-2c{)}^2+(s-2d{)}^2\\ & =(s^2-4as+4a^2)+(s^2-4bs+4b^2)+(s^2-4cs+4c^2)+(s^2-4ds+4d^2)\\ & =4s^2-4(a+b+c+d)s+4a^2+4b^2+4c^2+4d^2\\ & =4s^2-4s^2+4a^2+4b^2+4c^2+4d^2\\ & =4(a^2+b^2+c^2+d^2)\\ & =4(3+5+6+7)\\ & =\boxed{84}.\end{array}$$