Question 6
For a prime $p=4k+3$, where $k\in\mathbb Z_{>0}$, consider
$$\begin{aligned}Q(x)={}&px^{2p}-x^{2p-1}+p^2x^{(3p+1)/2}-px^{p+1}\\&+2(p^2+1)x^p-px^{p-1}\\&+p^2x^{(p-1)/2}-x+p.\end{aligned}$$
Find all ordered pairs of polynomials $f(x),g(x)$ with integer coefficients such that $Q(x)=f(x)g(x)$.
$$\begin{aligned}Q(x)={}&px^{2p}-x^{2p-1}+p^2x^{(3p+1)/2}-px^{p+1}\\&+2(p^2+1)x^p-px^{p-1}\\&+p^2x^{(p-1)/2}-x+p.\end{aligned}$$
Find all ordered pairs of polynomials $f(x),g(x)$ with integer coefficients such that $Q(x)=f(x)g(x)$.