<s>[INST] How to factorize this determinant? 
The question is to factorize $$\det\begin{pmatrix}(x^2+1)^2 & (xy+1)^2 & (xz+1)^2  \\ (xy+1)^2 & (y^2+1)^2 & (yz+1)^2 \\ (xz+1)^2 & (yz+1)^2 & (z^2+1)^2  \end{pmatrix}.$$

I have a hint which is considering the factorization of $\det\begin{pmatrix}1 & a & a^2  \\ 1 & b & b^2 \\ 1 & c & c^2  \end{pmatrix}$ where all entries are real numbers. 
I don't have any idea how to use the hint. So, I factorize $\det\begin{pmatrix}(x^2+1)^2 & (xy+1)^2 & (xz+1)^2  \\ (xy+1)^2 & (y^2+1)^2 & (yz+1)^2 \\ (xz+1)^2 & (yz+1)^2 & (z^2+1)^2  \end{pmatrix}$ directly and I get the answer which is $2(z-y)^2(z-x)^2(y-x)^2$.
My question is how to use the hint to factorize the given determinant? It is because my method seems very tedious.
 [/INST] By calculation (I got this slightly mixed up in my comment),
$$\begin{pmatrix} 1 & 2x & x^2 \\ 1 & 2y & y^2 \\ 1 & 2z & z^2 \end{pmatrix} \begin{pmatrix} 1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2 \end{pmatrix}^\top = \begin{pmatrix} 1 + 2x^2 + x^4 & 1 + 2xy + (xy)^2 & 1 + 2xz + (xz)^2 \\ 1 + 2xy + (xy)^2 & 1 + 2y^2 + y^4 & 1 + 2yz + (yz)^2 \\ 1 + 2xz + (xz)^2 & 1 + 2yz + (yz)^2 & 1 + 2z^2 + z^4 \end{pmatrix}.$$
Using the hint, the first matrix has determinant $2(x - y)(y - z)(x - z)$ and the second matrix has the determinant $(x - y)(y - z)(x - z)$. We can now just multiply these determinants together to get the determinant of the right hand side, which is the matrix in question.
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