Simplify $ \dfrac{x^2-1}{x^2+2x+1} $ to $ \dfrac{x-1}{x+1} $.

Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{x+1}$.

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

Divide $ \dfrac{x-1}{x+1} $ by $ 1-x $ to get $ \dfrac{ 1 }{ -x-1 } $.

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

Step 2: Factor numerators and denominators.

Step 3: Cancel common factors.

Step 4: Multiply numerators and denominators.

Step 5: Simplify numerator and denominator.

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

Divide $ \dfrac{1}{-x-1} $ by $ x+1 $ to get $ \dfrac{1}{-x^2-2x-1} $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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