Question
Answer
$$-(x+3)^2(x-2)^2(x-4)=-x^5+2x^4+19x^3-32x^2-84x+144$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}-(x+3)^2(x-2)^2(x-4)& \xlongequal{ }-(x^2+6x+9)(x^2-4x+4)(x-4) \xlongequal{ } \\[1 em] & \xlongequal{ }-(x^4+2x^3-11x^2-12x+36)(x-4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}-(x^5-2x^4-19x^3+32x^2+84x-144) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}-x^5+2x^4+19x^3-32x^2-84x+144\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x^4+2x^3-11x^2-12x+36}\right) $ by each term in $ \left( x-4\right) $. $$ \left( \color{blue}{x^4+2x^3-11x^2-12x+36}\right) \cdot \left( x-4\right) = x^5-4x^4+2x^4-8x^3-11x^3+44x^2-12x^2+48x+36x-144 $$ |
| ② | Combine like terms: $$ x^5 \color{blue}{-4x^4} + \color{blue}{2x^4} \color{red}{-8x^3} \color{red}{-11x^3} + \color{green}{44x^2} \color{green}{-12x^2} + \color{orange}{48x} + \color{orange}{36x} -144 = \\ = x^5 \color{blue}{-2x^4} \color{red}{-19x^3} + \color{green}{32x^2} + \color{orange}{84x} -144 $$ |
| ③ | Remove the parentheses by changing the sign of each term within them. $$ - \left(x^5-2x^4-19x^3+32x^2+84x-144 \right) = -x^5+2x^4+19x^3-32x^2-84x+144 $$ |
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