Answer
$$x(1+3x+2x^2)^2+(1+2x)(1+3x+x^2)=4x^5+12x^4+15x^3+13x^2+6x+1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}x(1+3x+2x^2)^2+(1+2x)(1+3x+x^2)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}x(4x^4+12x^3+13x^2+6x+1)+(1+2x)(1+3x+x^2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}4x^5+12x^4+13x^3+6x^2+x+1+3x+x^2+2x+6x^2+2x^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}4x^5+12x^4+13x^3+6x^2+x+2x^3+7x^2+5x+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}4x^5+12x^4+15x^3+13x^2+6x+1\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{1+3x+2x^2}\right) $ by each term in $ \left( 1+3x+2x^2\right) $. $$ \left( \color{blue}{1+3x+2x^2}\right) \cdot \left( 1+3x+2x^2\right) = 1+3x+2x^2+3x+9x^2+6x^3+2x^2+6x^3+4x^4 $$ |
| ② | Combine like terms: $$ 1+ \color{blue}{3x} + \color{red}{2x^2} + \color{blue}{3x} + \color{green}{9x^2} + \color{orange}{6x^3} + \color{green}{2x^2} + \color{orange}{6x^3} +4x^4 = \\ = 4x^4+ \color{orange}{12x^3} + \color{green}{13x^2} + \color{blue}{6x} +1 $$ |
| ③ | Multiply $ \color{blue}{x} $ by $ \left( 4x^4+12x^3+13x^2+6x+1\right) $ $$ \color{blue}{x} \cdot \left( 4x^4+12x^3+13x^2+6x+1\right) = 4x^5+12x^4+13x^3+6x^2+x $$ Multiply each term of $ \left( \color{blue}{1+2x}\right) $ by each term in $ \left( 1+3x+x^2\right) $. $$ \left( \color{blue}{1+2x}\right) \cdot \left( 1+3x+x^2\right) = 1+3x+x^2+2x+6x^2+2x^3 $$ |
| ④ | Combine like terms: $$ 1+ \color{blue}{3x} + \color{red}{x^2} + \color{blue}{2x} + \color{red}{6x^2} +2x^3 = 2x^3+ \color{red}{7x^2} + \color{blue}{5x} +1 $$ |
| ⑤ | Combine like terms: $$ 4x^5+12x^4+ \color{blue}{13x^3} + \color{red}{6x^2} + \color{green}{x} + \color{blue}{2x^3} + \color{red}{7x^2} + \color{green}{5x} +1 = \\ = 4x^5+12x^4+ \color{blue}{15x^3} + \color{red}{13x^2} + \color{green}{6x} +1 $$ |
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