$x^4-32x^2+1 = \left(x^2-\frac{16-\sqrt{252}}{4}\right)\left(x^2-\frac{16-\sqrt{252}}{4}\right)$
$=\left(x-\sqrt{\frac{16-\sqrt{252}}{4}}\right)\left(x+\sqrt{\frac{16-\sqrt{252}}{4}}\right)\left(x-\sqrt{\frac{16+\sqrt{252}}{4}}\right)\left(x+\sqrt{\frac{16+\sqrt{252}}{4}}\right)$