Let's break down the expression step by step:

\[ 2020 \times 2022 + \frac{2021}{2022} \times 2021 - 1 \]

First, calculate \( 2020 \times 2022 \):

\[ 2020 \times 2022 = 2020 \times (2020 + 2) = 2020 \times 2020 + 2020 \times 2 \]
\[ 2020 \times 2020 = 4080400 \]
\[ 2020 \times 2 = 4040 \]
\[ 4080400 + 4040 = 4084440 \]

Next, calculate \( \frac{2021}{2022} \times 2021 \):

\[ \frac{2021}{2022} \times 2021 = \frac{2021 \times 2021}{2022} \]
\[ 2021 \times 2021 = 4084441 \]
\[ \frac{4084441}{2022} \approx 2020.5 \]

Now, add the results together and subtract 1:

\[ 4084440 + 2020.5 - 1 \]
\[ 4084440 + 2020.5 = 4086460.5 \]
\[ 4086460.5 - 1 = 4086459.5 \]

So, the final result is:

\[ 4086459.5 \]