<s>[INST] Other ways to factorize $xy\left(x+y\right)+yz\left(y-z\right)-xz\left(x+z\right)$ To factorize $xy\left(x+y\right)+yz\left(y-z\right)-xz\left(x+z\right)$ I used the fact that $x=-y$ and $y=z$ and $x=-z$ make the expression zero. Hence it factorize to $\lambda (x+y)(y-z)(x+z)$ and we can check that the number $\lambda$ is equal to $1$.
I'm looking for other approaches/ideas to factorize the expression.
 [/INST] You can observe that, if $x=-y$ or (not "and") $x=-z$ or (not "and") $y=z$ , then the expression equals to zero. Thus, we have
$$xy(x+y)+yz(y-z)-xz(x+z)=A(x + y) (x + z) (y - z)$$
Here, $A$ must be a real number, not an expression. Because, it is enough to see that the coefficient of $x^2y$ equals to $1$. This implies, $A=1$.

Write,
$$A=(x+z)y^2+y(x^2-z^2)-xz(x+z)\\
\Delta_y=(x-z)^2(x+z)^2+4xz(x+z)^2=(x+z)^2(x+z)^2=(x+z)^4$$
Then we have,
$$y_{1,2}=\frac{(z^2-x^2)±(x+z)^2}{2(x+z)}$$
Can you take from here?
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