Answer
$$-(x+3)^2(x-2)^2(x+1)(x-1)=-x^6-2x^5+12x^4+14x^3-47x^2-12x+36$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}-(x+3)^2(x-2)^2(x+1)(x-1)& \xlongequal{ }-(x^2+6x+9)(x^2-4x+4)(x+1)(x-1) \xlongequal{ } \\[1 em] & \xlongequal{ }-(x^4+2x^3-11x^2-12x+36)(x+1)(x-1) \xlongequal{ } \\[1 em] & \xlongequal{ }-(x^5+3x^4-9x^3-23x^2+24x+36)(x-1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}-(x^6+2x^5-12x^4-14x^3+47x^2+12x-36) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}-x^6-2x^5+12x^4+14x^3-47x^2-12x+36\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x^5+3x^4-9x^3-23x^2+24x+36}\right) $ by each term in $ \left( x-1\right) $. $$ \left( \color{blue}{x^5+3x^4-9x^3-23x^2+24x+36}\right) \cdot \left( x-1\right) = \\ = x^6-x^5+3x^5-3x^4-9x^4+9x^3-23x^3+23x^2+24x^2-24x+36x-36 $$ |
| ② | Combine like terms: $$ x^6 \color{blue}{-x^5} + \color{blue}{3x^5} \color{red}{-3x^4} \color{red}{-9x^4} + \color{green}{9x^3} \color{green}{-23x^3} + \color{orange}{23x^2} + \color{orange}{24x^2} \color{blue}{-24x} + \color{blue}{36x} -36 = \\ = x^6+ \color{blue}{2x^5} \color{red}{-12x^4} \color{green}{-14x^3} + \color{orange}{47x^2} + \color{blue}{12x} -36 $$ |
| ③ | Remove the parentheses by changing the sign of each term within them. $$ - \left(x^6+2x^5-12x^4-14x^3+47x^2+12x-36 \right) = -x^6-2x^5+12x^4+14x^3-47x^2-12x+36 $$ |
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