Divide $ \dfrac{b+1}{b^2+14b+48} $ by $ b-1 $ to get $ \dfrac{b+1}{b^3+13b^2+34b-48} $.

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.

$$ \begin{aligned} \frac{ \frac{b+1}{b^2+14b+48} }{b-1} & \xlongequal{\text{Step 1}} \frac{b+1}{b^2+14b+48} \cdot \frac{\color{blue}{1}}{\color{blue}{b-1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( b+1 \right) \cdot 1 }{ \left( b^2+14b+48 \right) \cdot \left( b-1 \right) } \xlongequal{\text{Step 3}} \frac{ b+1 }{ b^3-b^2+14b^2-14b+48b-48 } = \\[1ex] &= \frac{b+1}{b^3+13b^2+34b-48} \end{aligned} $$

Divide $ \dfrac{b+1}{b^3+13b^2+34b-48} $ by $ b+6 $ to get $ \dfrac{b+1}{b^4+19b^3+112b^2+156b-288} $.

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.

$$ \begin{aligned} \frac{ \frac{b+1}{b^3+13b^2+34b-48} }{b+6} & \xlongequal{\text{Step 1}} \frac{b+1}{b^3+13b^2+34b-48} \cdot \frac{\color{blue}{1}}{\color{blue}{b+6}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( b+1 \right) \cdot 1 }{ \left( b^3+13b^2+34b-48 \right) \cdot \left( b+6 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ b+1 }{ b^4+6b^3+13b^3+78b^2+34b^2+204b-48b-288 } = \frac{b+1}{b^4+19b^3+112b^2+156b-288} \end{aligned} $$