1. determine how many, what type, and find the roots for f(x) = x4 + 21x2 − 100.

Note x^2 =y and rewrite this quartile equation in y and will get a quadratic what you need to solve it 

f(x) = x^4 +21x^2 -100 note x^2 =y

f(y) = y^2 +21y -100

y^2 +21y -100 = 0 

D = 21^2 -4(-100) = 441 +400 = 841

y_1,2 = (-21 +/- sqrt841 )/2 = (-21 +/- 29)/2 = -50/2 = - 25 and 8/2 = 4 

so y_1 = - 25 and y_2 = 4

x_1,2 = sqrt(y_1) = sqrt(-25) = +/- 5i

x_3,4 = sqrt4 = +/- 2

hope this will help you