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