Complex Roots of Polynomial

P(x) = x^4 + cx^2 + e, where c and e are real

If sqrt3 + i is a root, determine c and e, then factor into quadratic factors with linear factors

Here's my attempt:

Since c and e are real,

then sqrt3 - i is a root (complex conjugates theorem)

Also

P(-x) = P(x)

therefore P(x) is even

Hence

-(sqrt3 + i) and -(sqrt3 - i) are also roots

from there i got e = 16 (product of roots)

c= ...

However the answers have it differently.

It subbed sqrt3 + i and sqrt3 - i into P(x), then solved simultaneously acquiring:

e= - 8, c = 8

Can someone explain the flaw in my reasoning? Or perhaps answers is wrong?