Answer
$$x\cdot(1+2x)\cdot(1+3x)+(1+5x+4x^2)(1+4x+2x^2)=8x^4+32x^3+31x^2+10x+1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}x\cdot(1+2x)\cdot(1+3x)+(1+5x+4x^2)(1+4x+2x^2)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x+2x^2)\cdot(1+3x)+8x^4+26x^3+26x^2+9x+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}x+3x^2+2x^2+6x^3+8x^4+26x^3+26x^2+9x+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}6x^3+5x^2+x+8x^4+26x^3+26x^2+9x+1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}8x^4+32x^3+31x^2+10x+1\end{aligned} $$ | |
| ① | Multiply $ \color{blue}{x} $ by $ \left( 1+2x\right) $ $$ \color{blue}{x} \cdot \left( 1+2x\right) = x+2x^2 $$ Multiply each term of $ \left( \color{blue}{1+5x+4x^2}\right) $ by each term in $ \left( 1+4x+2x^2\right) $. $$ \left( \color{blue}{1+5x+4x^2}\right) \cdot \left( 1+4x+2x^2\right) = 1+4x+2x^2+5x+20x^2+10x^3+4x^2+16x^3+8x^4 $$ |
| ② | Combine like terms: $$ 1+ \color{blue}{4x} + \color{red}{2x^2} + \color{blue}{5x} + \color{green}{20x^2} + \color{orange}{10x^3} + \color{green}{4x^2} + \color{orange}{16x^3} +8x^4 = \\ = 8x^4+ \color{orange}{26x^3} + \color{green}{26x^2} + \color{blue}{9x} +1 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x+2x^2}\right) $ by each term in $ \left( 1+3x\right) $. $$ \left( \color{blue}{x+2x^2}\right) \cdot \left( 1+3x\right) = x+3x^2+2x^2+6x^3 $$ |
| ④ | Combine like terms: $$ x+ \color{blue}{3x^2} + \color{blue}{2x^2} +6x^3 = 6x^3+ \color{blue}{5x^2} +x $$ |
| ⑤ | Combine like terms: $$ \color{blue}{6x^3} + \color{red}{5x^2} + \color{green}{x} +8x^4+ \color{blue}{26x^3} + \color{red}{26x^2} + \color{green}{9x} +1 = 8x^4+ \color{blue}{32x^3} + \color{red}{31x^2} + \color{green}{10x} +1 $$ |
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