jmc

Expanding and completing the square, we get $$\begin{array}{rl}(12-x)(10-x)(12+x)(10+x)& =(10+x)(10-x)(12+x)(12-x)\\ & =(100-x^2)(144-x^2)\\ & =x^4-244x^2+14400\\ & =(x^2-122{)}^2-484.\end{array}$$The minimum value of $\boxed{-484}$ occurs at $x=\pm \sqrt{122}.$