parmenides51 wrote: Assume $n$ is a positive integer and integer $a$ is the root of the equation $$x^4+3ax^2+2ax-2\times 3^n=0.$$Find all $n$ and $ a$ that satisfy the conditions. $$(n,a)=(1,1),(2,-3)$$

We have $a^2(a^2+3a+2)=2\cdot 3^n$. Then $a$ is odd hence $a=\epsilon 3^k$ where $\epsilon=\pm 1$ and $k\ge 0$. Inserting this gives $3^{2k}(3^{2k}+\epsilon 3^{k+1}+2)=2\cdot 3^n$. If $k=0$ then $3(1+\epsilon)=2\cdot 3^n \implies \epsilon=1,n=1,a=1$. If $k\ge 1$ then $2k=n\implies 3^{k+1}(3^{k-1}+\epsilon)=0\implies k=1,\epsilon=-1,a=-3,n=2$.