Answer
$$(x+1)(x-4)(x-8)^3=x^5-27x^4+260x^3-992x^2+768x+2048$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x+1)(x-4)(x-8)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x+1)(x-4)(x^3-24x^2+192x-512) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^2-4x+x-4)(x^3-24x^2+192x-512) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^2-3x-4)(x^3-24x^2+192x-512) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^5-27x^4+260x^3-992x^2+768x+2048\end{aligned} $$ | |
| ① | Find $ \left(x-8\right)^3 $ using formula $$ (A - B) = A^3 - 3A^2B + 3AB^2 - B^3 $$where $ A = x $ and $ B = 8 $. $$ \left(x-8\right)^3 = x^3-3 \cdot x^2 \cdot 8 + 3 \cdot x \cdot 8^2-8^3 = x^3-24x^2+192x-512 $$ |
| ② | Multiply each term of $ \left( \color{blue}{x+1}\right) $ by each term in $ \left( x-4\right) $. $$ \left( \color{blue}{x+1}\right) \cdot \left( x-4\right) = x^2-4x+x-4 $$ |
| ③ | Combine like terms: $$ x^2 \color{blue}{-4x} + \color{blue}{x} -4 = x^2 \color{blue}{-3x} -4 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^2-3x-4}\right) $ by each term in $ \left( x^3-24x^2+192x-512\right) $. $$ \left( \color{blue}{x^2-3x-4}\right) \cdot \left( x^3-24x^2+192x-512\right) = \\ = x^5-24x^4+192x^3-512x^2-3x^4+72x^3-576x^2+1536x-4x^3+96x^2-768x+2048 $$ |
| ⑤ | Combine like terms: $$ x^5 \color{blue}{-24x^4} + \color{red}{192x^3} \color{green}{-512x^2} \color{blue}{-3x^4} + \color{orange}{72x^3} \color{blue}{-576x^2} + \color{red}{1536x} \color{orange}{-4x^3} + \color{blue}{96x^2} \color{red}{-768x} +2048 = \\ = x^5 \color{blue}{-27x^4} + \color{orange}{260x^3} \color{blue}{-992x^2} + \color{red}{768x} +2048 $$ |
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