Let's determine the values of $p$ and $q$ for which

$$
f(x) \equiv x^{4}+p x^{2}+q x+a^{2}
$$

is a multiple of $\left(x^{2}-1\right)$. What will be the other factor of $f(x)$ in this case?
The polynomial expression $x^{2}-1$ is a multiple if it is a multiple of both $(x-1)$ and $(x+1)$, meaning that the value of the polynomial is zero when $x=+1$ and $x=-1$. Therefore, it must be that:

$$
1+p+q+a^{2}=0 \quad \text { and } \quad 1+p-q+a^{2}=0
$$

From this:

$$
p=-\left(a^{2}+1\right), \quad q=0
$$

Accordingly, the polynomial expression is:

$$
x^{4}-\left(a^{2}+1\right) x^{3}+a^{2}=\left(x^{2}-1\right)\left(x^{2}-a^{2}\right)
$$

Dorn Tibor (Szent István g. V. o. Bp. XIV.)