What are all of the real roots of the following polynomial? f (x) = x4 - 24x2 - 25

It factors as ...

f (x) = (x² - 25) (x² + 1) = (x - 5) (x + 5) (x² + 1)

A root will make one of the factors equal to zero. The quadratic factor has only complex roots.

The real roots are x = ±5.