Add $b^4-8b^3+10b^2+2b$ and $ \dfrac{4}{b} $ to get $ \dfrac{ \color{purple}{ b^5-8b^4+10b^3+2b^2+4 } }{ b }$.

Step 1: Write $ b^4-8b^3+10b^2+2b $ 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 $\color{blue}{ b }$.

$$ \begin{aligned} b^4-8b^3+10b^2+2b+ \frac{4}{b} & \xlongequal{\text{Step 1}} \frac{b^4-8b^3+10b^2+2b}{\color{red}{1}} + \frac{4}{b} = \frac{ \left( b^4-8b^3+10b^2+2b \right) \cdot \color{blue}{ b }}{ 1 \cdot \color{blue}{ b }} + \frac{ 4 }{ b } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ b^5-8b^4+10b^3+2b^2 } }{ b } + \frac{ \color{purple}{ 4 } }{ b }=\frac{ \color{purple}{ b^5-8b^4+10b^3+2b^2+4 } }{ b } \end{aligned} $$

Subtract $2$ from $ \dfrac{b^5-8b^4+10b^3+2b^2+4}{b} $ to get $ \dfrac{ \color{purple}{ b^5-8b^4+10b^3+2b^2-2b+4 } }{ b }$.

Step 1: Write $ 2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{b}$.

$$ \begin{aligned} \frac{b^5-8b^4+10b^3+2b^2+4}{b} -2 & \xlongequal{\text{Step 1}} \frac{b^5-8b^4+10b^3+2b^2+4}{b} - \frac{2}{\color{red}{1}} = \frac{ b^5-8b^4+10b^3+2b^2+4 }{ b } - \frac{ 2 \cdot \color{blue}{ b }}{ 1 \cdot \color{blue}{ b }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ b^5-8b^4+10b^3+2b^2+4 } }{ b } - \frac{ \color{purple}{ 2b } }{ b }=\frac{ \color{purple}{ b^5-8b^4+10b^3+2b^2-2b+4 } }{ b } \end{aligned} $$