Monic Polynomial with absolute 1 as all coefficients

Question

Determine all monic polynomials $p(x)$ with integer coefficients of degree two for which there exists a polynomial $q(x)$ with integer coefficients such that $p(x)q(x)$ is a polynomial having all coefficients $\pm1$.

What I found is, suppose $p(x)=x^2+ax+b$. Then we get [after some not nice unfactorizing $p(x)q(x)$ and while putting the general polynomial form for $q(x)$, and using some relations] that

• $b=\pm1$
• $a=\pm1,0$

Hence $p(x)$ has $6$ possible values. Those are :

1. $p(x)=x^2\pm1$
2. $p(x)=x^2+x\pm1$
3. $p(x)=x^2-x\pm1$

So is my solution complete?