Problem 9.4. Find all integers $a$ such that the equation

$$
x^{4}+2 x^{3}+\left(a^{2}-a-9\right) x^{2}-4 x+4=0
$$

has at least one real root.

Stoyan Atanasov
9.4. Set $u=x-\frac{2}{x}$. Then the equation becomes

$$
u^{2}+2 u+a^{2}-a-5=0
$$

Since the equation $x^{2}-u x-2=0$ has real solutions for any real $u$, it suffices to find the integer values of $a$ for which the equation (1) has a real root. The last holds when $D=-a^{2}+a+6 \geq 0$, i.e. $(a-3)(a+2) \leq 0$ giving $a \in[-2,3]$. Therefore $a=-2,-1,0,1,2,3$.