Question
Answer
$$(s(s+2)+0.5ks(s+1))(0.1s+1)+k(s+1)=ks+s^2+k+2s$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(s(s+2)+0.5ks(s+1))(0.1s+1)+k(s+1)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(1s^2+2s+0ks(s+1))(0s+1)+ks+k \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(1s^2+2s+0ks^2+0ks)(0s+1)+ks+k \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(1s^2+2s)(0s+1)+ks+k \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}0s^3+s^2+0s^2+2s+ks+k \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}s^2+2s+ks+k \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}ks+s^2+k+2s\end{aligned} $$ | |
| 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 $ \color{blue}{0ks} $ by $ \left( s+1\right) $ $$ \color{blue}{0ks} \cdot \left( s+1\right) = 0ks^20ks $$ |
| ③ | Combine like terms: $$ s^2+2s0ks^20ks = s^2+2s $$ |
| ④ | Multiply each term of $ \left( \color{blue}{s^2+2s}\right) $ by each term in $ \left( 0s+1\right) $. $$ \left( \color{blue}{s^2+2s}\right) \cdot \left( 0s+1\right) = 0s^3+s^20s^2+2s $$ |
| ⑤ | Combine like terms: $$ 0s^3+ \color{blue}{s^2} \color{blue}{0s^2} +2s = \color{blue}{s^2} +2s $$ |
| ⑥ | Combine like terms: $$ s^2+2s+ks+k = ks+s^2+k+2s $$ |
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