Question

Is it possible for a degree-4 polynomial P(x) to have only complex (no real) roots

Answer 1

work backwards from 4 imaginary roots (remember they come in complex conjugate pairs)
(x-i)(x+i)(x-2i)(x+2i) = 0
(x^2+1)(x^2+4)= 0
x^4 + 5 x^2 + 4 = 0

Answer 2

Is it possible for a degree-4 polynomial P(x) to have only complex (no real) roots?