Answer
$$(3-x)\cdot(3-x)\cdot(4-x)-8\cdot(3-x)-9\cdot(4-x)+24=-x^3+10x^2-16x$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(3-x)\cdot(3-x)\cdot(4-x)-8\cdot(3-x)-9\cdot(4-x)+24& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(9-3x-3x+x^2)\cdot(4-x)-(24-8x)-(36-9x)+24 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^2-6x+9)\cdot(4-x)-(24-8x)-(36-9x)+24 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}4x^2-x^3-24x+6x^2+36-9x-(24-8x)-(36-9x)+24 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-x^3+10x^2-33x+36-(24-8x)-(36-9x)+24 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-x^3+10x^2-33x+36-24+8x-(36-9x)+24 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}-x^3+10x^2-25x+12-(36-9x)+24 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}-x^3+10x^2-25x+12-36+9x+24 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}-x^3+10x^2-16x\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{3-x}\right) $ by each term in $ \left( 3-x\right) $. $$ \left( \color{blue}{3-x}\right) \cdot \left( 3-x\right) = 9-3x-3x+x^2 $$Multiply $ \color{blue}{8} $ by $ \left( 3-x\right) $ $$ \color{blue}{8} \cdot \left( 3-x\right) = 24-8x $$Multiply $ \color{blue}{9} $ by $ \left( 4-x\right) $ $$ \color{blue}{9} \cdot \left( 4-x\right) = 36-9x $$ |
| ② | Combine like terms: $$ 9 \color{blue}{-3x} \color{blue}{-3x} +x^2 = x^2 \color{blue}{-6x} +9 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^2-6x+9}\right) $ by each term in $ \left( 4-x\right) $. $$ \left( \color{blue}{x^2-6x+9}\right) \cdot \left( 4-x\right) = 4x^2-x^3-24x+6x^2+36-9x $$ |
| ④ | Combine like terms: $$ \color{blue}{4x^2} -x^3 \color{red}{-24x} + \color{blue}{6x^2} +36 \color{red}{-9x} = -x^3+ \color{blue}{10x^2} \color{red}{-33x} +36 $$ |
| ⑤ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 24-8x \right) = -24+8x $$ |
| ⑥ | Combine like terms: $$ -x^3+10x^2 \color{blue}{-33x} + \color{red}{36} \color{red}{-24} + \color{blue}{8x} = -x^3+10x^2 \color{blue}{-25x} + \color{red}{12} $$ |
| ⑦ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 36-9x \right) = -36+9x $$ |
| ⑧ | Combine like terms: $$ -x^3+10x^2 \color{blue}{-25x} + \color{red}{12} \color{green}{-36} + \color{blue}{9x} + \color{green}{24} = -x^3+10x^2 \color{blue}{-16x} $$ |
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