$$ \left( \color{blue}{ip+1}\right) \cdot x = ipx+x $$

Multiply $p$ by $ \dfrac{i}{2} $ to get $ \dfrac{ ip }{ 2 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} p \cdot \frac{i}{2} & \xlongequal{\text{Step 1}} \frac{p}{\color{red}{1}} \cdot \frac{i}{2} \xlongequal{\text{Step 2}} \frac{ p \cdot i }{ 1 \cdot 2 } \xlongequal{\text{Step 3}} \frac{ ip }{ 2 } \end{aligned} $$

Remove the parentheses by changing the sign of each term within them.

$$ - \left( ipx+x \right) = -ipx-x $$

Add $ \dfrac{-i^2p}{2} $ and $ \dfrac{ip}{2} $ to get $ \dfrac{ -i^2p + ip }{ \color{blue}{ 2 }}$.

To add expressions with the same denominators, we add the numerators and write the result over the common denominator.

$$ \begin{aligned} \frac{-i^2p}{2} + \frac{ip}{2} & = \frac{-i^2p}{\color{blue}{2}} + \frac{ip}{\color{blue}{2}} =\frac{ -i^2p + ip }{ \color{blue}{ 2 }} \end{aligned} $$

Divide $x^2-ipx-x+ip$ by $ \dfrac{-i^2p+ip}{2} $ to get $ \dfrac{-2ipx+2ip+2x^2-2x}{-i^2p+ip} $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

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

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} \frac{x^2-ipx-x+ip}{ \frac{\color{blue}{-i^2p+ip}}{\color{blue}{2}} } & \xlongequal{\text{Step 1}} x^2-ipx-x+ip \cdot \frac{\color{blue}{2}}{\color{blue}{-i^2p+ip}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{x^2-ipx-x+ip}{\color{red}{1}} \cdot \frac{2}{-i^2p+ip} \xlongequal{\text{Step 3}} \frac{ \left( x^2-ipx-x+ip \right) \cdot 2 }{ 1 \cdot \left( -i^2p+ip \right) } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ 2x^2-2ipx-2x+2ip }{ -i^2p+ip } = \frac{-2ipx+2ip+2x^2-2x}{-i^2p+ip} \end{aligned} $$