Answer
$$(x^2+(x^2-x^3)(3x-2x^2+x^3))(2x-x^2)=x^8-5x^7+11x^6-13x^5+5x^4+2x^3$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x^2+(x^2-x^3)(3x-2x^2+x^3))(2x-x^2)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2+3x^3-2x^4+x^5-3x^4+2x^5-x^6)(2x-x^2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^2-x^6+3x^5-5x^4+3x^3)(2x-x^2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(-x^6+3x^5-5x^4+3x^3+x^2)(2x-x^2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^8-5x^7+11x^6-13x^5+5x^4+2x^3\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x^2-x^3}\right) $ by each term in $ \left( 3x-2x^2+x^3\right) $. $$ \left( \color{blue}{x^2-x^3}\right) \cdot \left( 3x-2x^2+x^3\right) = 3x^3-2x^4+x^5-3x^4+2x^5-x^6 $$ |
| ② | Combine like terms: $$ 3x^3 \color{blue}{-2x^4} + \color{red}{x^5} \color{blue}{-3x^4} + \color{red}{2x^5} -x^6 = -x^6+ \color{red}{3x^5} \color{blue}{-5x^4} +3x^3 $$ |
| ③ | Combine like terms: $$ x^2-x^6+3x^5-5x^4+3x^3 = -x^6+3x^5-5x^4+3x^3+x^2 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{-x^6+3x^5-5x^4+3x^3+x^2}\right) $ by each term in $ \left( 2x-x^2\right) $. $$ \left( \color{blue}{-x^6+3x^5-5x^4+3x^3+x^2}\right) \cdot \left( 2x-x^2\right) = -2x^7+x^8+6x^6-3x^7-10x^5+5x^6+6x^4-3x^5+2x^3-x^4 $$ |
| ⑤ | Combine like terms: $$ \color{blue}{-2x^7} +x^8+ \color{red}{6x^6} \color{blue}{-3x^7} \color{green}{-10x^5} + \color{red}{5x^6} + \color{orange}{6x^4} \color{green}{-3x^5} +2x^3 \color{orange}{-x^4} = \\ = x^8 \color{blue}{-5x^7} + \color{red}{11x^6} \color{green}{-13x^5} + \color{orange}{5x^4} +2x^3 $$ |
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