Multiply $ \dfrac{1-x^2}{1+y} $ by $ \dfrac{1-y^2}{x+x^2} $ to get $ \dfrac{ xy-x-y+1 }{ x } $.

Step 1: Factor numerators and denominators.

Step 2: Cancel common factors.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

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

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

Step 1: Write $ 1 $ 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 $\color{blue}{ -x+1 }$.

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

Multiply $ \dfrac{xy-x-y+1}{x} $ by $ \dfrac{1}{-x+1} $ to get $ \dfrac{ xy-x-y+1 }{ -x^2+x } $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

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