Answer
$$(x-6)(x-7)(x-8)(x-9)=x^4-30x^3+335x^2-1650x+3024$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-6)(x-7)(x-8)(x-9)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2-7x-6x+42)(x-8)(x-9) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^2-13x+42)(x-8)(x-9) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3-8x^2-13x^2+104x+42x-336)(x-9) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(x^3-21x^2+146x-336)(x-9) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}x^4-30x^3+335x^2-1650x+3024\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x-6}\right) $ by each term in $ \left( x-7\right) $. $$ \left( \color{blue}{x-6}\right) \cdot \left( x-7\right) = x^2-7x-6x+42 $$ |
| ② | Combine like terms: $$ x^2 \color{blue}{-7x} \color{blue}{-6x} +42 = x^2 \color{blue}{-13x} +42 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^2-13x+42}\right) $ by each term in $ \left( x-8\right) $. $$ \left( \color{blue}{x^2-13x+42}\right) \cdot \left( x-8\right) = x^3-8x^2-13x^2+104x+42x-336 $$ |
| ④ | Combine like terms: $$ x^3 \color{blue}{-8x^2} \color{blue}{-13x^2} + \color{red}{104x} + \color{red}{42x} -336 = x^3 \color{blue}{-21x^2} + \color{red}{146x} -336 $$ |
| ⑤ | Multiply each term of $ \left( \color{blue}{x^3-21x^2+146x-336}\right) $ by each term in $ \left( x-9\right) $. $$ \left( \color{blue}{x^3-21x^2+146x-336}\right) \cdot \left( x-9\right) = x^4-9x^3-21x^3+189x^2+146x^2-1314x-336x+3024 $$ |
| ⑥ | Combine like terms: $$ x^4 \color{blue}{-9x^3} \color{blue}{-21x^3} + \color{red}{189x^2} + \color{red}{146x^2} \color{green}{-1314x} \color{green}{-336x} +3024 = \\ = x^4 \color{blue}{-30x^3} + \color{red}{335x^2} \color{green}{-1650x} +3024 $$ |
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