## Task B-4.2.

Determine $n \in \mathbb{N}$ such that the polynomial $P(x)=\left(2 x^{2}+1\right)^{n+1}-(3 x+15)^{2 n}$ is divisible by the polynomial $Q(x)=x+2$. In this case, determine all the non-real roots of the polynomial $P$.
## Solution.

A polynomial $P$ is divisible by $x+2$ if and only if $P(-2)=0$.

It holds that $P(-2)=9^{n+1}-9^{2 n}$, so from $9^{n+1}-9^{2 n}=0$ it follows that $n+1=2 n$, or $n=1$.

Then the polynomial $P$ is

$$
P(x)=\left(2 x^{2}+1\right)^{2}-(3 x+15)^{2}=\left(2 x^{2}-3 x-14\right)\left(2 x^{2}+3 x+16\right)
$$

The sought roots are the solutions to the equation $2 x^{2}+3 x+16=0$, 1 point

and they are $x_{1,2}=\frac{-3 \pm i \sqrt{119}}{4}$.

1 point