This is my follow-up question to my own query earlier:
Almost half of the answers said that I provided my own proof by giving the counterexample. Although, I was not satisfied through those answers since they didn't provide a complete view of the situation and how there exist algebraic factors.
Is disproving an algebraic statement necessarily the same as introducing a case where it is not true? I agree with Nameless's statement found here, but I am not sure as to how that contributes to a rigorous algebraic proof.
Remark: I agree that I didn't ask for an algebraic proof earlier, but I did expect a better proof.