Question
Answer
$$(x-3)(x+2)^2(x+1)=x^4+2x^3-7x^2-20x-12$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-3)(x+2)^2(x+1)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x-3)(x^2+4x+4)(x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^3+4x^2+4x-3x^2-12x-12)(x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3+x^2-8x-12)(x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^4+2x^3-7x^2-20x-12\end{aligned} $$ | |
| 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) $. $$ \left( \color{blue}{x^3+x^2-8x-12}\right) \cdot \left( x+1\right) = x^4+x^3+x^3+x^2-8x^2-8x-12x-12 $$ |
| ⑤ | Combine like terms: $$ x^4+ \color{blue}{x^3} + \color{blue}{x^3} + \color{red}{x^2} \color{red}{-8x^2} \color{green}{-8x} \color{green}{-12x} -12 = x^4+ \color{blue}{2x^3} \color{red}{-7x^2} \color{green}{-20x} -12 $$ |
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