Question
Answer
$$(x+2)^2(x-7)(x-12)=x^4-15x^3+12x^2+260x+336$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x+2)^2(x-7)(x-12)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2+4x+4)(x-7)(x-12) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^3-7x^2+4x^2-28x+4x-28)(x-12) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3-3x^2-24x-28)(x-12) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^4-15x^3+12x^2+260x+336\end{aligned} $$ | |
| 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^2+4x+4}\right) $ by each term in $ \left( x-7\right) $. $$ \left( \color{blue}{x^2+4x+4}\right) \cdot \left( x-7\right) = x^3-7x^2+4x^2-28x+4x-28 $$ |
| ③ | Combine like terms: $$ x^3 \color{blue}{-7x^2} + \color{blue}{4x^2} \color{red}{-28x} + \color{red}{4x} -28 = x^3 \color{blue}{-3x^2} \color{red}{-24x} -28 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^3-3x^2-24x-28}\right) $ by each term in $ \left( x-12\right) $. $$ \left( \color{blue}{x^3-3x^2-24x-28}\right) \cdot \left( x-12\right) = x^4-12x^3-3x^3+36x^2-24x^2+288x-28x+336 $$ |
| ⑤ | Combine like terms: $$ x^4 \color{blue}{-12x^3} \color{blue}{-3x^3} + \color{red}{36x^2} \color{red}{-24x^2} + \color{green}{288x} \color{green}{-28x} +336 = \\ = x^4 \color{blue}{-15x^3} + \color{red}{12x^2} + \color{green}{260x} +336 $$ |
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