The value of $$\frac{1}{\left(9 - 4\sqrt{5}\right)^2} + \frac{1}{\left(9 + 4\sqrt{5}\right)^2}$$ is:
$$\frac{1}{\left(9 - 4\sqrt{5}\right)^2} + \frac{1}{\left(9 + 4\sqrt{5}\right)^2}=\frac{1}{81+80-72\sqrt{5}}+\frac{1}{81+80+72\sqrt{5}}$$
$$=\frac{1}{161-72\sqrt{5}}+\frac{1}{161+72\sqrt{5}}$$
$$=\frac{161+72\sqrt{5}+161-72\sqrt{5}}{161^2-\left(72\sqrt{5}\right)^2}$$
$$=\frac{322}{25921-25920}$$
$$=322$$
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