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