$$(s(s+2)+\dfrac{1}{2}ks(s+1))(\dfrac{1}{10}s+1)+k(s+1)=\dfrac{ks^3+11ks^2+2s^3+30ks+24s^2+20k+40s}{20}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(s(s+2)+\frac{1}{2}ks(s+1))(\frac{1}{10}s+1)+k(s+1)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(1s^2+2s+\frac{1}{2}ks(s+1))(\frac{1}{10}s+1)+ks+k \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(1s^2+2s+\frac{k}{2}s(s+1))(\frac{s}{10}+1)+ks+k \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}(1s^2+2s+\frac{ks}{2}(s+1))\frac{s+10}{10}+ks+k \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}(1s^2+2s+\frac{ks^2+ks}{2})\frac{s+10}{10}+ks+k \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}\frac{ks^2+ks+2s^2+4s}{2}\frac{s+10}{10}+ks+k \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}\frac{ks^3+11ks^2+2s^3+10ks+24s^2+40s}{20}+ks+k \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle11}{\textcircled {11}} } }}}\frac{ks^3+11ks^2+2s^3+30ks+24s^2+20k+40s}{20}\end{aligned} $$ | |
| ① | Multiply $ \color{blue}{s} $ by $ \left( s+2\right) $ $$ \color{blue}{s} \cdot \left( s+2\right) = s^2+2s $$Multiply $ \color{blue}{k} $ by $ \left( s+1\right) $ $$ \color{blue}{k} \cdot \left( s+1\right) = ks+k $$ |
| ② | 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. $$ \begin{aligned} \frac{1}{2} \cdot k & \xlongequal{\text{Step 1}} \frac{1}{2} \cdot \frac{k}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot k }{ 2 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ k }{ 2 } \end{aligned} $$ |
| ③ | 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. $$ \begin{aligned} \frac{1}{10} \cdot s & \xlongequal{\text{Step 1}} \frac{1}{10} \cdot \frac{s}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot s }{ 10 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ s }{ 10 } \end{aligned} $$ |
| ④ | 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. $$ \begin{aligned} \frac{k}{2} \cdot s & \xlongequal{\text{Step 1}} \frac{k}{2} \cdot \frac{s}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ k \cdot s }{ 2 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ ks }{ 2 } \end{aligned} $$ |
| ⑤ | 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. |
| ⑥ | 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. $$ \begin{aligned} \frac{ks}{2} \cdot s+1 & \xlongequal{\text{Step 1}} \frac{ks}{2} \cdot \frac{s+1}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ ks \cdot \left( s+1 \right) }{ 2 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ ks^2+ks }{ 2 } \end{aligned} $$ |
| ⑦ | 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. |
| ⑧ | Add $s^2+2s$ and $ \dfrac{ks^2+ks}{2} $ to get $ \dfrac{ \color{purple}{ ks^2+ks+2s^2+4s } }{ 2 }$. Step 1: Write $ s^2+2s $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: To add raitonal expressions, both fractions must have the same denominator. |
| ⑨ | 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. |
| ⑩ | Multiply $ \dfrac{ks^2+ks+2s^2+4s}{2} $ by $ \dfrac{s+10}{10} $ to get $ \dfrac{ks^3+11ks^2+2s^3+10ks+24s^2+40s}{20} $. Step 1: Multiply numerators and denominators. Step 2: Simplify numerator and denominator. $$ \begin{aligned} \frac{ks^2+ks+2s^2+4s}{2} \cdot \frac{s+10}{10} & \xlongequal{\text{Step 1}} \frac{ \left( ks^2+ks+2s^2+4s \right) \cdot \left( s+10 \right) }{ 2 \cdot 10 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ ks^3+10ks^2+ks^2+10ks+2s^3+20s^2+4s^2+40s }{ 20 } = \frac{ks^3+11ks^2+2s^3+10ks+24s^2+40s}{20} \end{aligned} $$ |
| ⑪ | Add $ \dfrac{ks^3+11ks^2+2s^3+10ks+24s^2+40s}{20} $ and $ ks+k $ to get $ \dfrac{ \color{purple}{ ks^3+11ks^2+2s^3+30ks+24s^2+20k+40s } }{ 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. |