Exercise
Suppose
R = Q[X]/(d),
where d = X4 + aX + 1.
For which a 
Q is
Q}
a subring of R?
All a
a = 0
a = 1
No a
No:
X2 S,
but
X2 * X2 =
X4 = -aX-1
does not belong to S in general.
S,
but
X2 * X2 =
X4 = -aX-1
does not belong to S in general.
Yes.
No:
X2 S,
but
X2 * X2 =
X4 = -X-1
does not belong to S.
S,
but
X2 * X2 =
X4 = -X-1
does not belong to S.
No, check your argument.