## Task B-2.1.

Determine the smallest and largest integer $x$ for which the number $x^{4}-2021 x^{2}+2020$ is negative.
## Solution.

We are looking for solutions to the inequality $x^{4}-2021 x^{2}+2020<0$.

Let's factorize the expression on the left side.

$$
\begin{array}{cc}
x^{4}-2020 x^{2}-x^{2}+2020<0 & 1 \text { point } \\
x^{2}\left(x^{2}-2020\right)-\left(x^{2}-2020\right)<0 & \\
\left(x^{2}-2020\right)\left(x^{2}-1\right)<0 & 1 \text { point }
\end{array}
$$

From here, we have $1<x^{2}<2020$, which implies $1<|x|<\sqrt{2020}$, or

$$
x \in\langle-\sqrt{2020},-1\rangle \cup\langle 1, \sqrt{2020}\rangle
$$

Since $44=\sqrt{1936}<\sqrt{2020}<\sqrt{2025}=45$, the smallest integer for which the given expression is negative is -44, and the largest is 44.