Answer: 373

Small Hint:

Use Sophie Germain’s identity with $324=4\cdot3^4$

Big Hint:

If $g(x)=x^2+6x+18,$ rewrite $x^4+324$ as $g(x-6)g(x)$

Solution:

Let $g(x)=x^2+6x+18.$ Sophie Germain’s identity gives $x^4+324=g(x-6)g(x).$ The numerator therefore supplies $g(4),g(10),\ldots,g(58),$ while the denominator supplies $g(-2),g(4),\ldots,g(52).$ Everything cancels except $$\begin{aligned}\frac{g(58)}{g(-2)}&=\frac{58^2+6(58)+18}{4-12+18}\\&=\frac{3730}{10}=373.\end{aligned}$$