1. Find all monic integer polynomials $P(x)$ of degree 2 such that there exists an integer polynomial $Q(x)$ for which all coefficients of $P(x) Q(x)$ are $\pm 1$.
1. $P(x)=x^{2} \pm x \pm 1, x^{2} \pm 1, x^{2} \pm 2 x+1$.

Let $F(x)$ be any polynomial of degree $n$ with coefficients all being $\pm 1$. If $z$ is a root of $F(x)$, and $|z|>1$, then
$$
\begin{array}{l}
|z|^{n}=\left|z^{n}\right|=\left| \pm z^{n-1} \pm z^{n-2} \pm \cdots \pm 1\right| \\
\leqslant|z|^{n-1}+|z|^{n-2}+\cdots+1=\frac{|z|^{n}-1}{|z|-1} .
\end{array}
$$

Thus, we have $|z|^{n}(|z|-2) \leqslant-1$.
Therefore, $|z|<2$.
Hence, all roots of $F(x)$ have a modulus less than 2.
Let $P(x)=x^{2}+a x \pm 1$, where $a$ is an integer, and $x_{1}, x_{2}$ are its two roots, then $x_{1} x_{2}= \pm 1$.
Without loss of generality, assume $\left|x_{1}\right| \geqslant 1,\left|x_{2}\right| \leqslant 1$.
Since $x_{1}, x_{2}$ are also roots of $P(x) Q(x)$, and the coefficients of $P(x) Q(x)$ are all $\pm 1$, we have $\left|x_{1}\right|<2$.
Thus, $|a|=\left|x_{1}+x_{2}\right| \leqslant\left|x_{1}\right|+\left|x_{2}\right|<3$, i.e., $a \in\{ \pm 2, \pm 1,0\}$.
When $a= \pm 1$, take $Q(x)=1$;
When $a=0$, take $Q(x)=x+1$;
When $a= \pm 2$, the polynomial $x^{2} \pm 2 x-1$ has a root with an absolute value greater than 2, which does not meet the problem's requirements. If $P(x)=x^{2} \pm 2 x+1$, take $Q(x)=x \mp 1$.