The real part (one side) must equal the real part of the other side, and the coefficients of the imaginary parts of both sides must be equal. this gives us
a + c + (b + d)i = 0 + 0i -> a + c = 0 and b + d = 0 -> c=-a d=-b

the additive inverse of z was defined as as c+di. –> F = -a +(-b)i -> F=-a-bi