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