Question
Answer
$$(x+2)(x+1)(x-1)(x-5)=x^4-3x^3-11x^2+3x+10$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x+2)(x+1)(x-1)(x-5)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2+x+2x+2)(x-1)(x-5) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^2+3x+2)(x-1)(x-5) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3-x^2+3x^2-3x+2x-2)(x-5) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(x^3+2x^2-x-2)(x-5) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}x^4-3x^3-11x^2+3x+10\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x+2}\right) $ by each term in $ \left( x+1\right) $. $$ \left( \color{blue}{x+2}\right) \cdot \left( x+1\right) = x^2+x+2x+2 $$ |
| ② | Combine like terms: $$ x^2+ \color{blue}{x} + \color{blue}{2x} +2 = x^2+ \color{blue}{3x} +2 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^2+3x+2}\right) $ by each term in $ \left( x-1\right) $. $$ \left( \color{blue}{x^2+3x+2}\right) \cdot \left( x-1\right) = x^3-x^2+3x^2-3x+2x-2 $$ |
| ④ | Combine like terms: $$ x^3 \color{blue}{-x^2} + \color{blue}{3x^2} \color{red}{-3x} + \color{red}{2x} -2 = x^3+ \color{blue}{2x^2} \color{red}{-x} -2 $$ |
| ⑤ | Multiply each term of $ \left( \color{blue}{x^3+2x^2-x-2}\right) $ by each term in $ \left( x-5\right) $. $$ \left( \color{blue}{x^3+2x^2-x-2}\right) \cdot \left( x-5\right) = x^4-5x^3+2x^3-10x^2-x^2+5x-2x+10 $$ |
| ⑥ | Combine like terms: $$ x^4 \color{blue}{-5x^3} + \color{blue}{2x^3} \color{red}{-10x^2} \color{red}{-x^2} + \color{green}{5x} \color{green}{-2x} +10 = x^4 \color{blue}{-3x^3} \color{red}{-11x^2} + \color{green}{3x} +10 $$ |
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