Multiply $ \color{blue}{k} $ by $ \left( s+1\right) $ $$ \color{blue}{k} \cdot \left( s+1\right) = ks+k $$

Multiply $ \dfrac{1}{2} $ by $ k $ to get $ \dfrac{ k }{ 2 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{2} \cdot k & \xlongequal{\text{Step 1}} \frac{1}{2} \cdot \frac{k}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot k }{ 2 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ k }{ 2 } \end{aligned} $$

Multiply $ \dfrac{1}{10} $ by $ s $ to get $ \dfrac{ s }{ 10 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{10} \cdot s & \xlongequal{\text{Step 1}} \frac{1}{10} \cdot \frac{s}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot s }{ 10 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ s }{ 10 } \end{aligned} $$

Multiply $ \dfrac{k}{2} $ by $ s $ to get $ \dfrac{ ks }{ 2 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{k}{2} \cdot s & \xlongequal{\text{Step 1}} \frac{k}{2} \cdot \frac{s}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ k \cdot s }{ 2 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ ks }{ 2 } \end{aligned} $$

Add $ \dfrac{s}{10} $ and $ 1 $ to get $ \dfrac{ \color{purple}{ s+10 } }{ 10 }$.

Step 1: Write $ 1 $ 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 second fraction by $\color{blue}{10}$.

$$ \begin{aligned} \frac{s}{10} +1 & \xlongequal{\text{Step 1}} \frac{s}{10} + \frac{1}{\color{red}{1}} = \frac{ s }{ 10 } + \frac{ 1 \cdot \color{blue}{ 10 }}{ 1 \cdot \color{blue}{ 10 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ s } }{ 10 } + \frac{ \color{purple}{ 10 } }{ 10 }=\frac{ \color{purple}{ s+10 } }{ 10 } \end{aligned} $$

Multiply $ \dfrac{ks}{2} $ by $ s+1 $ to get $ \dfrac{ ks^2+ks }{ 2 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $ \dfrac{s}{10} $ and $ 1 $ to get $ \dfrac{ \color{purple}{ s+10 } }{ 10 }$.

Step 1: Write $ 1 $ 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 second fraction by $\color{blue}{10}$.

$$ \begin{aligned} \frac{s}{10} +1 & \xlongequal{\text{Step 1}} \frac{s}{10} + \frac{1}{\color{red}{1}} = \frac{ s }{ 10 } + \frac{ 1 \cdot \color{blue}{ 10 }}{ 1 \cdot \color{blue}{ 10 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ s } }{ 10 } + \frac{ \color{purple}{ 10 } }{ 10 }=\frac{ \color{purple}{ s+10 } }{ 10 } \end{aligned} $$

Add $s+2$ and $ \dfrac{ks^2+ks}{2} $ to get $ \dfrac{ \color{purple}{ ks^2+ks+2s+4 } }{ 2 }$.

Step 1: Write $ s+2 $ 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 $\color{blue}{ 2 }$.

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

Add $ \dfrac{s}{10} $ and $ 1 $ to get $ \dfrac{ \color{purple}{ s+10 } }{ 10 }$.

Step 1: Write $ 1 $ 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 second fraction by $\color{blue}{10}$.

$$ \begin{aligned} \frac{s}{10} +1 & \xlongequal{\text{Step 1}} \frac{s}{10} + \frac{1}{\color{red}{1}} = \frac{ s }{ 10 } + \frac{ 1 \cdot \color{blue}{ 10 }}{ 1 \cdot \color{blue}{ 10 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ s } }{ 10 } + \frac{ \color{purple}{ 10 } }{ 10 }=\frac{ \color{purple}{ s+10 } }{ 10 } \end{aligned} $$

Multiply $ \dfrac{ks^2+ks+2s+4}{2} $ by $ \dfrac{s+10}{10} $ to get $ \dfrac{ks^3+11ks^2+10ks+2s^2+24s+40}{20} $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ks^2+ks+2s+4}{2} \cdot \frac{s+10}{10} & \xlongequal{\text{Step 1}} \frac{ \left( ks^2+ks+2s+4 \right) \cdot \left( s+10 \right) }{ 2 \cdot 10 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ ks^3+10ks^2+ks^2+10ks+2s^2+20s+4s+40 }{ 20 } = \frac{ks^3+11ks^2+10ks+2s^2+24s+40}{20} \end{aligned} $$

Add $ \dfrac{ks^3+11ks^2+10ks+2s^2+24s+40}{20} $ and $ ks+k $ to get $ \dfrac{ \color{purple}{ ks^3+11ks^2+30ks+2s^2+20k+24s+40 } }{ 20 }$.

Step 1: Write $ ks+k $ 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 second fraction by $\color{blue}{20}$.

$$ \begin{aligned} \frac{ks^3+11ks^2+10ks+2s^2+24s+40}{20} +ks+k & \xlongequal{\text{Step 1}} \frac{ks^3+11ks^2+10ks+2s^2+24s+40}{20} + \frac{ks+k}{\color{red}{1}} = \\[1ex] &= \frac{ ks^3+11ks^2+10ks+2s^2+24s+40 }{ 20 } + \frac{ \left( ks+k \right) \cdot \color{blue}{ 20 }}{ 1 \cdot \color{blue}{ 20 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ ks^3+11ks^2+10ks+2s^2+24s+40 } }{ 20 } + \frac{ \color{purple}{ 20ks+20k } }{ 20 } = \\[1ex] &=\frac{ \color{purple}{ ks^3+11ks^2+30ks+2s^2+20k+24s+40 } }{ 20 } \end{aligned} $$