jmc

Each of the terms is of the form $x^4+324$. To factor, we write: $$\begin{array}{rl}{x}^4+324& =(x^4+36x^2+324)-36x^2\\ & =(x^2+18{)}^2-36x^2\\ & =(x^2-6x+18)(x^2+6x+18)\\ & =(x(x-6)+18)(x(x+6)+18).\end{array}$$Therefore, the given expression equals $$\frac{(10\cdot 4+18)(10\cdot 16+18)(22\cdot 16+18)(22\cdot 28+18)\cdots (58\cdot 52+18)(58\cdot 64+18)}{(4\cdot (-2)+18)(4\cdot 10+18)(16\cdot 10+18)(16\cdot 22+18)\cdots (52\cdot 46+18)(52\cdot 58+18)}.$$Nearly all the terms cancel, leaving just $$\frac{58\cdot 64+18}{4\cdot (-2)+18}=\boxed{373}.$$Remark. The factorization $x^4+324=(x^2-6x+18)(x^2+6x+18)$ is a special case of the Sophie Germain identity, which is derived in the same way; it states that $$a^4+4b^4=(a^2-2ab+2b^2)(a^2+2ab+2b^2).$$