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