jmc

Write $a=x^2-50x-10$ and $b=x^2+25x+5$. Then the equation given becomes $$\frac{a+2b-1}{2}=ab,$$so $0=2ab-a-2b+1=(a-1)(2b-1)$. Then $a-1=x^2-50x-11=0$ or $2b-1=2x^2+50x+9=0$. The former has a positive root, $x=\boxed{25+2\sqrt{159}}$, while the latter does not.