<s>[INST] $a,b,c\in \Bbb Z$ and $a\cdot b\cdot c$ is a root of $ax^2+bx+c$. I was curious if there are quadratic equations where $a,b,c\in \Bbb Z$ and $a\cdot b\cdot c$ is a root of $ax^2+bx+c$.
So trivially if $c=0$, $a$ and $b$ can be arbitrary, and if either $a$ or $b$ is zero, this implies that $c=0$, and the other arbitrary.
Is there a way to find other solutions, if any exist?
So the equation to solve in integers is $$a^3b^2c^2+ab^2c+c=0.$$
Dividing out by $c$ gives $$a^3b^2c+ab^2+1=0.$$
We could factor to give $$ab^2(a^2c+1)=-1$$
So either 


*

*$ab^2=1$ and $a^2c+1=-1$  or

*$ab^2=-1$ and $a^2c+1=1$


For 1. $a=1$ and $b=\pm1$ and $c=-2$
For 2. $a=-1$ and $b=\pm1$ and $c=-2$ and $c=0$ is forced.
Is this analysis correct?
Are the non-trivial solutions:
$x^2+x-2$ and $x^2-x-2$, with roots $-2$ and $2$ respectively?
 [/INST] The proof looks correct to me. 
However, it might be good to say "However, we assumed $c \neq 0$" after point 2. 
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