Question

Compute $\frac{({10}^4+324)({22}^4+324)({34}^4+324)({46}^4+324)({58}^4+324)}{(4^4+324)({16}^4+324)({28}^4+324)({40}^4+324)({52}^4+324)}$

Answer 1

$x^4+324=\left[{\left(x-3\right)}^2+9\right]\left[{\left(x+3\right)}^2+9\right]$

from using above formula

$\begin{array}{c}\frac{\left(7^2+9\right)\left({13}^2+9\right)\left({19}^2+9\right)\left({25}^2+9\right)\left({31}^2+9\right)\left({37}^2+9\right)\left({43}^2+9\right)\left({55}^2+9\right)\left({61}^2+9\right)}{\left(1^2+9\right)\left(7^2+9\right)\left({13}^2+9\right)\left({19}^2+9\right)\left({25}^2+9\right)\left({31}^2+9\right)\left({37}^2+9\right)\left({43}^2+9\right)\left({55}^2+9\right)}\\ \frac{\left({61}^2+9\right)}{\left(1^2+9\right)}=\frac{3730}{10}=373\end{array}$