I do this problem in the homework paper and I think both of $(x+1)(x+3)$ and $(x-1)(x-3)$ are rational can imply $(x+1)(x+3)-(x-1)(x-3)=8x$ is rational and then $(x+1)^2$ is rational. But the answer shows both of $(x+1)(x+3)$ and $(x-1)(x-3)$ are rational cannot imply $(x+1)^2$ is rational. I am not sure whether me or the answer is incorrect. Can you help me?
Edit: Recheck the question, and I am sorry that I ignore one condition "x is irrational", so the answer is correct. Noticed in fact that condition 1 and 2 cannot hold at the same time.
Question: x is irrational, then can imply $(x+1)^2$ is irrational.
Condition 1: $(x+1)(x+3)$ is rational
condition 2: $(x-1)(x-3)$ is rational
Choose:
A: condition 1 is sufficient
B: condition 2 is sufficient
C: none of condition 1 or 2 is sufficient but condition 1 + 2 is sufficient
D: both of condition 1 and 2 is sufficient
E: none of condition 1 or 2 is sufficient and condition 1+2 is insufficient