How do you find all the roots of #f(x) = x^4 + 2x^3 + x^2 - 2x - 2#?
This quartic polynomial has zeros
First note that the sum of the coefficients is
#x^4+2x^3+x^2-2x-2 = (x-1)(x^3+3x^2+4x+2)#
If you reverse the signs of the coefficients of the terms of odd degree in the remaining cubic
#x^3+3x^2+4x+2 = (x+1)(x^2+2x+2)#
The remaining quadratic factor has negative discriminant, but you can factor it by completing the square with Complex coefficients:
#x^2+2x+2 = x^2+2x+1+1 = (x+1)^2-i^2 = (x+1-i)(x+1+i)#