Answer
$$(x+3)(x-2)^2(x+1)=x^4-9x^2+4x+12$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x+3)(x-2)^2(x+1)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x+3)(x^2-4x+4)(x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^3-4x^2+4x+3x^2-12x+12)(x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3-x^2-8x+12)(x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^4-9x^2+4x+12\end{aligned} $$ | |
| ① | Find $ \left(x-2\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}{ 2 }$. $$ \begin{aligned}\left(x-2\right)^2 = \color{blue}{x^2} -2 \cdot x \cdot 2 + \color{red}{2^2} = x^2-4x+4\end{aligned} $$ |
| ② | Multiply each term of $ \left( \color{blue}{x+3}\right) $ by each term in $ \left( x^2-4x+4\right) $. $$ \left( \color{blue}{x+3}\right) \cdot \left( x^2-4x+4\right) = x^3-4x^2+4x+3x^2-12x+12 $$ |
| ③ | Combine like terms: $$ x^3 \color{blue}{-4x^2} + \color{red}{4x} + \color{blue}{3x^2} \color{red}{-12x} +12 = x^3 \color{blue}{-x^2} \color{red}{-8x} +12 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^3-x^2-8x+12}\right) $ by each term in $ \left( x+1\right) $. $$ \left( \color{blue}{x^3-x^2-8x+12}\right) \cdot \left( x+1\right) = x^4+ \cancel{x^3} -\cancel{x^3}-x^2-8x^2-8x+12x+12 $$ |
| ⑤ | Combine like terms: $$ x^4+ \, \color{blue}{ \cancel{x^3}} \, \, \color{blue}{ -\cancel{x^3}} \, \color{green}{-x^2} \color{green}{-8x^2} \color{orange}{-8x} + \color{orange}{12x} +12 = x^4 \color{green}{-9x^2} + \color{orange}{4x} +12 $$ |
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