Problem 1078

Solution:
The equation has two distinct positive roots if and only if
$$\left\lvert\, \begin{aligned} & D=a^{3}-a^{2}-9 a+9>0 \\ & x_{1}+x_{2}=\frac{6}{a^{2}-a-9}>0 \\ & x_{1} x_{2}=\frac{-a}{a^{2}-a-9}>0 \end{aligned} \right.$$
The first inequality is satisfied for $a \in(-3,1) \cup(3,+\infty)$, the second one for $a \in((1-\sqrt{37}) / 2,(1+\sqrt{37}) / 2)$, and the third - for every $a \in(-\infty, 0)$. Therefore the required values of $a$ are $a \in(-3,(1-\sqrt{37}) / 2)$.