Hello, Soroban,
I'm not quite certain if I remember correctly, but I believe that there exist a "coefficient theorem of Vieta" (that's the name I learned long, long time ago) which says:
If then
Let r be the missing root of the equation and using this theorem I get:
a = -(1 + 2 + 3 + r) = -6 - r which is zero here: thus r = -6
b = 1*2 + 1*3 + 1*r + 2*3 + 2*r + 3*r = 11 + 6r
c = -(1*2*3 + 1*2*r + 2*3*r) = -6 - 8r
d = (-1)^4 * (1*2*3*r) = 6r
According to my results b + d (these are your a+c) = 11+6r + 6r = 11 +12 r = 11 - 72 = -61
But - as I mentioned before - I'm not sure that this attempt is correct.
EB