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

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

Divide $ \dfrac{x}{x^2+2x+1} $ by $ x+1 $ to get $ \dfrac{x}{x^3+3x^2+3x+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.

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