Multiply $2$ by $ \dfrac{x}{z+1} $ to get $ \dfrac{ 2x }{ z+1 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 2 \cdot \frac{x}{z+1} & \xlongequal{\text{Step 1}} \frac{2}{\color{red}{1}} \cdot \frac{x}{z+1} \xlongequal{\text{Step 2}} \frac{ 2 \cdot x }{ 1 \cdot \left( z+1 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 2x }{ z+1 } \end{aligned} $$

Add $ \dfrac{2x}{z+1} $ and $ \dfrac{x^2}{x^2-1} $ to get $ \dfrac{ \color{purple}{ 2x^3+x^2z+x^2-2x } }{ x^2z+x^2-z-1 }$.

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}{ x^2-1 }$ and the second by $\color{blue}{ z+1 }$.

$$ \begin{aligned} \frac{2x}{z+1} + \frac{x^2}{x^2-1} & = \frac{ 2x \cdot \color{blue}{ \left( x^2-1 \right) }}{ \left( z+1 \right) \cdot \color{blue}{ \left( x^2-1 \right) }} + \frac{ x^2 \cdot \color{blue}{ \left( z+1 \right) }}{ \left( x^2-1 \right) \cdot \color{blue}{ \left( z+1 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 2x^3-2x } }{ x^2z-z+x^2-1 } + \frac{ \color{purple}{ x^2z+x^2 } }{ x^2z-z+x^2-1 } = \\[1ex] &=\frac{ \color{purple}{ 2x^3+x^2z+x^2-2x } }{ x^2z+x^2-z-1 } \end{aligned} $$

Multiply $ \dfrac{2x^3+x^2z+x^2-2x}{x^2z+x^2-z-1} $ by $ \dfrac{1-x}{x^3} $ to get $ \dfrac{-2x^4-x^3z+x^3+x^2z+3x^2-2x}{x^5z+x^5-x^3z-x^3} $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{2x^3+x^2z+x^2-2x}{x^2z+x^2-z-1} \cdot \frac{1-x}{x^3} & \xlongequal{\text{Step 1}} \frac{ \left( 2x^3+x^2z+x^2-2x \right) \cdot \left( 1-x \right) }{ \left( x^2z+x^2-z-1 \right) \cdot x^3 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 2x^3-2x^4+x^2z-x^3z+x^2-x^3-2x+2x^2 }{ x^5z+x^5-x^3z-x^3 } = \frac{-2x^4-x^3z+x^3+x^2z+3x^2-2x}{x^5z+x^5-x^3z-x^3} \end{aligned} $$