# Factorizing (x−1)(x−3)(x−5)(x−7)−64

Factorizing $\left(x-1\right)\left(x-3\right)\left(x-5\right)\left(x-7\right)-64$
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iljovskint
$\left(x-1\right)\left(x-7\right)=y-9$ and $\left(x-3\right)\left(x-5\right)=y-1$ for $y={\left(x-4\right)}^{2}$ leading to a factorization of $\left(y-1\right)\left(y-9\right)-64={y}^{2}-10y-55$
Here:
${y}^{2}-10y-55=\left(y-5-4\sqrt{5}\right)\left(y-5+4\sqrt{5}\right)=\left({\left(x-4\right)}^{2}-5-4\sqrt{5}\right)\left({\left(x-4\right)}^{2}-5+4\sqrt{5}\right)=\left(x-4+\sqrt{5+4\sqrt{5}}\right)\left(x-4-\sqrt{5+4\sqrt{5}}\right)\left(x-4+i\sqrt{4\sqrt{5}-5}\right)\left(x-4-i\sqrt{4\sqrt{5}-5}\right)$