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