[ Identical Transformations ]

What should $p$ and $q$ be so that the equality $A x^{4}+B x^{2}+C=A\left(x^{2}+p x+q\right)\left(x^{2}-p x+q\right)$ holds?
The right-hand side equals $A\left(\left(x^{2}+q\right)^{2}-(p x)^{2}\right)=A x^{4}+A\left(2 q-p^{2}\right) x^{2}+A q^{2}$. Therefore, $p$ and $q$ must be solutions to the system

$A\left(2 q-p^{2}\right)=B, A q^{2}=C$. The system has solutions (one to four), if $A$ and $C$ have the same sign and