Question
Answer
$$(s+1)(s+2)(s+3)(s^2+2s+2)=s^5+8s^4+25s^3+40s^2+34s+12$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(s+1)(s+2)(s+3)(s^2+2s+2)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(1s^2+2s+s+2)(s+3)(s^2+2s+2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(1s^2+3s+2)(s+3)(s^2+2s+2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(1s^3+3s^2+3s^2+9s+2s+6)(s^2+2s+2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(1s^3+6s^2+11s+6)(s^2+2s+2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}s^5+8s^4+25s^3+40s^2+34s+12\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{s+1}\right) $ by each term in $ \left( s+2\right) $. $$ \left( \color{blue}{s+1}\right) \cdot \left( s+2\right) = s^2+2s+s+2 $$ |
| ② | Combine like terms: $$ s^2+ \color{blue}{2s} + \color{blue}{s} +2 = s^2+ \color{blue}{3s} +2 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{s^2+3s+2}\right) $ by each term in $ \left( s+3\right) $. $$ \left( \color{blue}{s^2+3s+2}\right) \cdot \left( s+3\right) = s^3+3s^2+3s^2+9s+2s+6 $$ |
| ④ | Combine like terms: $$ s^3+ \color{blue}{3s^2} + \color{blue}{3s^2} + \color{red}{9s} + \color{red}{2s} +6 = s^3+ \color{blue}{6s^2} + \color{red}{11s} +6 $$ |
| ⑤ | Multiply each term of $ \left( \color{blue}{s^3+6s^2+11s+6}\right) $ by each term in $ \left( s^2+2s+2\right) $. $$ \left( \color{blue}{s^3+6s^2+11s+6}\right) \cdot \left( s^2+2s+2\right) = \\ = s^5+2s^4+2s^3+6s^4+12s^3+12s^2+11s^3+22s^2+22s+6s^2+12s+12 $$ |
| ⑥ | Combine like terms: $$ s^5+ \color{blue}{2s^4} + \color{red}{2s^3} + \color{blue}{6s^4} + \color{green}{12s^3} + \color{orange}{12s^2} + \color{green}{11s^3} + \color{blue}{22s^2} + \color{red}{22s} + \color{blue}{6s^2} + \color{red}{12s} +12 = \\ = s^5+ \color{blue}{8s^4} + \color{green}{25s^3} + \color{blue}{40s^2} + \color{red}{34s} +12 $$ |
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