Add $4s^3+52s^2+228s+416$ and $ \dfrac{312}{s} $ to get $ \dfrac{ \color{purple}{ 4s^4+52s^3+228s^2+416s+312 } }{ s }$.

Step 1: Write $ 4s^3+52s^2+228s+416 $ 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 $\color{blue}{ s }$.

$$ \begin{aligned} 4s^3+52s^2+228s+416+ \frac{312}{s} & \xlongequal{\text{Step 1}} \frac{4s^3+52s^2+228s+416}{\color{red}{1}} + \frac{312}{s} = \frac{ \left( 4s^3+52s^2+228s+416 \right) \cdot \color{blue}{ s }}{ 1 \cdot \color{blue}{ s }} + \frac{ 312 }{ s } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 4s^4+52s^3+228s^2+416s } }{ s } + \frac{ \color{purple}{ 312 } }{ s }=\frac{ \color{purple}{ 4s^4+52s^3+228s^2+416s+312 } }{ s } \end{aligned} $$

Add $ \dfrac{4s^4+52s^3+228s^2+416s+312}{s} $ and $ \dfrac{80}{s^2} $ to get $ \dfrac{ \color{purple}{ 4s^6+52s^5+228s^4+416s^3+312s^2+80s } }{ s^3 }$.

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}{ s^2 }$ and the second by $\color{blue}{ s }$.

$$ \begin{aligned} \frac{4s^4+52s^3+228s^2+416s+312}{s} + \frac{80}{s^2} & = \frac{ \left( 4s^4+52s^3+228s^2+416s+312 \right) \cdot \color{blue}{ s^2 }}{ s \cdot \color{blue}{ s^2 }} + \frac{ 80 \cdot \color{blue}{ s }}{ s^2 \cdot \color{blue}{ s }} = \\[1ex] &=\frac{ \color{purple}{ 4s^6+52s^5+228s^4+416s^3+312s^2 } }{ s^3 } + \frac{ \color{purple}{ 80s } }{ s^3 } = \\[1ex] &=\frac{ \color{purple}{ 4s^6+52s^5+228s^4+416s^3+312s^2+80s } }{ s^3 } \end{aligned} $$

Subtract $s^3$ from $ \dfrac{4s^6+52s^5+228s^4+416s^3+312s^2+80s}{s^3} $ to get $ \dfrac{ \color{purple}{ 3s^6+52s^5+228s^4+416s^3+312s^2+80s } }{ s^3 }$.

Step 1: Write $ s^3 $ 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 $\color{blue}{s^3}$.

$$ \begin{aligned} \frac{4s^6+52s^5+228s^4+416s^3+312s^2+80s}{s^3} -s^3 & \xlongequal{\text{Step 1}} \frac{4s^6+52s^5+228s^4+416s^3+312s^2+80s}{s^3} - \frac{s^3}{\color{red}{1}} = \\[1ex] &= \frac{ 4s^6+52s^5+228s^4+416s^3+312s^2+80s }{ s^3 } - \frac{ s^3 \cdot \color{blue}{ s^3 }}{ 1 \cdot \color{blue}{ s^3 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 4s^6+52s^5+228s^4+416s^3+312s^2+80s } }{ s^3 } - \frac{ \color{purple}{ s^6 } }{ s^3 } = \\[1ex] &=\frac{ \color{purple}{ 3s^6+52s^5+228s^4+416s^3+312s^2+80s } }{ s^3 } \end{aligned} $$

Subtract $11s^2$ from $ \dfrac{3s^6+52s^5+228s^4+416s^3+312s^2+80s}{s^3} $ to get $ \dfrac{ \color{purple}{ 3s^6+41s^5+228s^4+416s^3+312s^2+80s } }{ s^3 }$.

Step 1: Write $ 11s^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 $\color{blue}{s^3}$.

$$ \begin{aligned} \frac{3s^6+52s^5+228s^4+416s^3+312s^2+80s}{s^3} -11s^2 & \xlongequal{\text{Step 1}} \frac{3s^6+52s^5+228s^4+416s^3+312s^2+80s}{s^3} - \frac{11s^2}{\color{red}{1}} = \\[1ex] &= \frac{ 3s^6+52s^5+228s^4+416s^3+312s^2+80s }{ s^3 } - \frac{ 11s^2 \cdot \color{blue}{ s^3 }}{ 1 \cdot \color{blue}{ s^3 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 3s^6+52s^5+228s^4+416s^3+312s^2+80s } }{ s^3 } - \frac{ \color{purple}{ 11s^5 } }{ s^3 } = \\[1ex] &=\frac{ \color{purple}{ 3s^6+41s^5+228s^4+416s^3+312s^2+80s } }{ s^3 } \end{aligned} $$

Subtract $37s$ from $ \dfrac{3s^6+41s^5+228s^4+416s^3+312s^2+80s}{s^3} $ to get $ \dfrac{ \color{purple}{ 3s^6+41s^5+191s^4+416s^3+312s^2+80s } }{ s^3 }$.

Step 1: Write $ 37s $ 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 $\color{blue}{s^3}$.

$$ \begin{aligned} \frac{3s^6+41s^5+228s^4+416s^3+312s^2+80s}{s^3} -37s & \xlongequal{\text{Step 1}} \frac{3s^6+41s^5+228s^4+416s^3+312s^2+80s}{s^3} - \frac{37s}{\color{red}{1}} = \\[1ex] &= \frac{ 3s^6+41s^5+228s^4+416s^3+312s^2+80s }{ s^3 } - \frac{ 37s \cdot \color{blue}{ s^3 }}{ 1 \cdot \color{blue}{ s^3 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 3s^6+41s^5+228s^4+416s^3+312s^2+80s } }{ s^3 } - \frac{ \color{purple}{ 37s^4 } }{ s^3 } = \\[1ex] &=\frac{ \color{purple}{ 3s^6+41s^5+191s^4+416s^3+312s^2+80s } }{ s^3 } \end{aligned} $$

Subtract $48$ from $ \dfrac{3s^6+41s^5+191s^4+416s^3+312s^2+80s}{s^3} $ to get $ \dfrac{ \color{purple}{ 3s^6+41s^5+191s^4+368s^3+312s^2+80s } }{ s^3 }$.

Step 1: Write $ 48 $ 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 $\color{blue}{s^3}$.

$$ \begin{aligned} \frac{3s^6+41s^5+191s^4+416s^3+312s^2+80s}{s^3} -48 & \xlongequal{\text{Step 1}} \frac{3s^6+41s^5+191s^4+416s^3+312s^2+80s}{s^3} - \frac{48}{\color{red}{1}} = \\[1ex] &= \frac{ 3s^6+41s^5+191s^4+416s^3+312s^2+80s }{ s^3 } - \frac{ 48 \cdot \color{blue}{ s^3 }}{ 1 \cdot \color{blue}{ s^3 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 3s^6+41s^5+191s^4+416s^3+312s^2+80s } }{ s^3 } - \frac{ \color{purple}{ 48s^3 } }{ s^3 } = \\[1ex] &=\frac{ \color{purple}{ 3s^6+41s^5+191s^4+368s^3+312s^2+80s } }{ s^3 } \end{aligned} $$

Subtract $ \dfrac{20}{s} $ from $ \dfrac{3s^6+41s^5+191s^4+368s^3+312s^2+80s}{s^3} $ to get $ \dfrac{ \color{purple}{ 3s^7+41s^6+191s^5+368s^4+292s^3+80s^2 } }{ s^4 }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ s }$ and the second by $\color{blue}{ s^3 }$.

$$ \begin{aligned} \frac{3s^6+41s^5+191s^4+368s^3+312s^2+80s}{s^3} - \frac{20}{s} & = \frac{ \left( 3s^6+41s^5+191s^4+368s^3+312s^2+80s \right) \cdot \color{blue}{ s }}{ s^3 \cdot \color{blue}{ s }} - \frac{ 20 \cdot \color{blue}{ s^3 }}{ s \cdot \color{blue}{ s^3 }} = \\[1ex] &=\frac{ \color{purple}{ 3s^7+41s^6+191s^5+368s^4+312s^3+80s^2 } }{ s^4 } - \frac{ \color{purple}{ 20s^3 } }{ s^4 } = \\[1ex] &=\frac{ \color{purple}{ 3s^7+41s^6+191s^5+368s^4+292s^3+80s^2 } }{ s^4 } \end{aligned} $$