Question
Answer
$$(x+3)^2(x-2)(x+1)=x^4+5x^3+x^2-21x-18$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x+3)^2(x-2)(x+1)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2+6x+9)(x-2)(x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^3-2x^2+6x^2-12x+9x-18)(x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3+4x^2-3x-18)(x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^4+5x^3+x^2-21x-18\end{aligned} $$ | |
| ① | Find $ \left(x+3\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}{ 3 }$. $$ \begin{aligned}\left(x+3\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 3 + \color{red}{3^2} = x^2+6x+9\end{aligned} $$ |
| ② | Multiply each term of $ \left( \color{blue}{x^2+6x+9}\right) $ by each term in $ \left( x-2\right) $. $$ \left( \color{blue}{x^2+6x+9}\right) \cdot \left( x-2\right) = x^3-2x^2+6x^2-12x+9x-18 $$ |
| ③ | Combine like terms: $$ x^3 \color{blue}{-2x^2} + \color{blue}{6x^2} \color{red}{-12x} + \color{red}{9x} -18 = x^3+ \color{blue}{4x^2} \color{red}{-3x} -18 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^3+4x^2-3x-18}\right) $ by each term in $ \left( x+1\right) $. $$ \left( \color{blue}{x^3+4x^2-3x-18}\right) \cdot \left( x+1\right) = x^4+x^3+4x^3+4x^2-3x^2-3x-18x-18 $$ |
| ⑤ | Combine like terms: $$ x^4+ \color{blue}{x^3} + \color{blue}{4x^3} + \color{red}{4x^2} \color{red}{-3x^2} \color{green}{-3x} \color{green}{-18x} -18 = x^4+ \color{blue}{5x^3} + \color{red}{x^2} \color{green}{-21x} -18 $$ |
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