$2.351 A=\frac{x^{8}+x^{4}-2 x^{2}+6}{x^{4}+2 x^{2}+3}+2 x^{2}-2$.
Solution. Let's divide the polynomial $x^{8}+x^{4}-2 x^{6}+6$ by the polynomial $x^{4}+2 x^{2}+3$.

![](https://cdn.mathpix.com/cropped/2024_05_21_f024bf2ff7725246f3bfg-034.jpg?height=163&width=525&top_left_y=502&top_left_x=89)

$$
\begin{aligned}
& \begin{array}{r}
-\frac{-2 x^{6}-4 x^{4}-6 x^{2}}{-2 x^{4}+4 x^{2}+6} \\
\frac{2 x^{4}+4 x^{2}+6}{0}
\end{array}
\end{aligned}
$$

Therefore, $A=\frac{\left(x^{4}+2 x^{2}+3\right)\left(x^{4}-2 x^{2}+2\right)}{x^{4}+2 x^{2}+3}+2 x^{2}-2=x^{4}$.

Answer: $A=x^{4}$.