Determine how many, what type, and find the roots for f(x) = x^4 + 21x^2 -100.

Among the coefficient signs (++-) there is one sign change, so Descarte's rule of signs tells you there is one positive real root. Changing x to -x does not affect the signs, so there is also one negative real root. Then there are two complex roots to bring the total number to 4, the degree of the polynomial.

The factorization is
  f(x) = (x^2 +25)(x^2 -4)
Each of these can be factored using the form for factoring the difference of squares.
  f(x) = (x -5i)(x +5i)(x -2)(x +2)

There are 2 real roots: ±2.
There are 2 imaginary roots: ±5i.