Multiply each term of $ \left( \color{blue}{x-2024}\right) $ by each term in $ \left( x+2023\right) $.

$$ \left( \color{blue}{x-2024}\right) \cdot \left( x+2023\right) = x^2+2023x-2024x-4094552 $$

Combine like terms:

$$ x^2+ \color{blue}{2023x} \color{blue}{-2024x} -4094552 = x^2 \color{blue}{-x} -4094552 $$

Subtract $ \dfrac{1}{2024} $ from $ x $ to get $ \dfrac{ \color{purple}{ 2024x-1 } }{ 2024 }$.

Step 1: Write $ x $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ 2024 }$.

$$ \begin{aligned} x- \frac{1}{2024} & \xlongequal{\text{Step 1}} \frac{x}{\color{red}{1}} - \frac{1}{2024} = \frac{ x \cdot \color{blue}{ 2024 }}{ 1 \cdot \color{blue}{ 2024 }} - \frac{ 1 }{ 2024 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2024x } }{ 2024 } - \frac{ \color{purple}{ 1 } }{ 2024 }=\frac{ \color{purple}{ 2024x-1 } }{ 2024 } \end{aligned} $$

Add $x$ and $ \dfrac{1}{2023} $ to get $ \dfrac{ \color{purple}{ 2023x+1 } }{ 2023 }$.

Step 1: Write $ x $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ 2023 }$.

$$ \begin{aligned} x+ \frac{1}{2023} & \xlongequal{\text{Step 1}} \frac{x}{\color{red}{1}} + \frac{1}{2023} = \frac{ x \cdot \color{blue}{ 2023 }}{ 1 \cdot \color{blue}{ 2023 }} + \frac{ 1 }{ 2023 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2023x } }{ 2023 } + \frac{ \color{purple}{ 1 } }{ 2023 }=\frac{ \color{purple}{ 2023x+1 } }{ 2023 } \end{aligned} $$

Multiply $x^2-x-4094552$ by $ \dfrac{2024x-1}{2024} $ to get $ \dfrac{2024x^3-2025x^2-8287373247x+4094552}{2024} $.

Step 1: Write $ x^2-x-4094552 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} x^2-x-4094552 \cdot \frac{2024x-1}{2024} & \xlongequal{\text{Step 1}} \frac{x^2-x-4094552}{\color{red}{1}} \cdot \frac{2024x-1}{2024} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x^2-x-4094552 \right) \cdot \left( 2024x-1 \right) }{ 1 \cdot 2024 } \xlongequal{\text{Step 3}} \frac{ 2024x^3-x^2-2024x^2+x-8287373248x+4094552 }{ 2024 } = \\[1ex] &= \frac{2024x^3-2025x^2-8287373247x+4094552}{2024} \end{aligned} $$

Add $x$ and $ \dfrac{1}{2023} $ to get $ \dfrac{ \color{purple}{ 2023x+1 } }{ 2023 }$.

Step 1: Write $ x $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ 2023 }$.

$$ \begin{aligned} x+ \frac{1}{2023} & \xlongequal{\text{Step 1}} \frac{x}{\color{red}{1}} + \frac{1}{2023} = \frac{ x \cdot \color{blue}{ 2023 }}{ 1 \cdot \color{blue}{ 2023 }} + \frac{ 1 }{ 2023 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2023x } }{ 2023 } + \frac{ \color{purple}{ 1 } }{ 2023 }=\frac{ \color{purple}{ 2023x+1 } }{ 2023 } \end{aligned} $$

Multiply $ \dfrac{2024x^3-2025x^2-x+4094552}{2024} $ by $ \dfrac{2023x+1}{2023} $ to get $ \dfrac{4094552x^4-4094551x^3-4048x^2+8283278695x+4094552}{4094552} $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{2024x^3-2025x^2-x+4094552}{2024} \cdot \frac{2023x+1}{2023} & \xlongequal{\text{Step 1}} \frac{ \left( 2024x^3-2025x^2-x+4094552 \right) \cdot \left( 2023x+1 \right) }{ 2024 \cdot 2023 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 4094552x^4+2024x^3-4096575x^3-2025x^2-2023x^2-x+8283278696x+4094552 }{ 4094552 } = \frac{4094552x^4-4094551x^3-4048x^2+8283278695x+4094552}{4094552} \end{aligned} $$