Question
Answer
$$2(x+1)^2(x-2)^2=2x^4-4x^3-6x^2+8x+8$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}2(x+1)^2(x-2)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}2(x^2+2x+1)(x^2-4x+4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(2x^2+4x+2)(x^2-4x+4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}2x^4-4x^3-6x^2+8x+8\end{aligned} $$ | |
| 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 }$. $$ \begin{aligned}\left(x+1\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 1 + \color{red}{1^2} = x^2+2x+1\end{aligned} $$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 $ \color{blue}{2} $ by $ \left( x^2+2x+1\right) $ $$ \color{blue}{2} \cdot \left( x^2+2x+1\right) = 2x^2+4x+2 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{2x^2+4x+2}\right) $ by each term in $ \left( x^2-4x+4\right) $. $$ \left( \color{blue}{2x^2+4x+2}\right) \cdot \left( x^2-4x+4\right) = 2x^4-8x^3+8x^2+4x^3-16x^2+16x+2x^2-8x+8 $$ |
| ④ | Combine like terms: $$ 2x^4 \color{blue}{-8x^3} + \color{red}{8x^2} + \color{blue}{4x^3} \color{green}{-16x^2} + \color{orange}{16x} + \color{green}{2x^2} \color{orange}{-8x} +8 = 2x^4 \color{blue}{-4x^3} \color{green}{-6x^2} + \color{orange}{8x} +8 $$ |
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