Question
Answer
$$672x^4+(2x+2)^3-2(112x-1)(2-40x+3x^3)=14x^3+8984x^2-504x+12$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}672x^4+(2x+2)^3-2(112x-1)(2-40x+3x^3)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}672x^4+8x^3+24x^2+24x+8-2(112x-1)(2-40x+3x^3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}672x^4+8x^3+24x^2+24x+8-(224x-2)(2-40x+3x^3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}672x^4+8x^3+24x^2+24x+8-(448x-8960x^2+672x^4-4+80x-6x^3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}672x^4+8x^3+24x^2+24x+8-(672x^4-6x^3-8960x^2+528x-4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}672x^4+8x^3+24x^2+24x+8-672x^4+6x^3+8960x^2-528x+4 \xlongequal{ } \\[1 em] & \xlongequal{ } \cancel{672x^4}+8x^3+24x^2+24x+8 -\cancel{672x^4}+6x^3+8960x^2-528x+4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}14x^3+8984x^2-504x+12\end{aligned} $$ | |
| ① | Find $ \left(2x+2\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = 2x $ and $ B = 2 $. $$ \left(2x+2\right)^3 = \left( 2x \right)^3+3 \cdot \left( 2x \right)^2 \cdot 2 + 3 \cdot 2x \cdot 2^2+2^3 = 8x^3+24x^2+24x+8 $$ |
| ② | Multiply $ \color{blue}{2} $ by $ \left( 112x-1\right) $ $$ \color{blue}{2} \cdot \left( 112x-1\right) = 224x-2 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{224x-2}\right) $ by each term in $ \left( 2-40x+3x^3\right) $. $$ \left( \color{blue}{224x-2}\right) \cdot \left( 2-40x+3x^3\right) = 448x-8960x^2+672x^4-4+80x-6x^3 $$ |
| ④ | Combine like terms: $$ 672x^4+8x^3+24x^2+24x+8 = 672x^4+8x^3+24x^2+24x+8 $$ |
| ⑤ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 672x^4-6x^3-8960x^2+528x-4 \right) = -672x^4+6x^3+8960x^2-528x+4 $$ |
| ⑥ | Combine like terms: $$ \, \color{blue}{ \cancel{672x^4}} \,+ \color{green}{8x^3} + \color{orange}{24x^2} + \color{blue}{24x} + \color{red}{8} \, \color{blue}{ -\cancel{672x^4}} \,+ \color{green}{6x^3} + \color{orange}{8960x^2} \color{blue}{-528x} + \color{red}{4} = \\ = \color{green}{14x^3} + \color{orange}{8984x^2} \color{blue}{-504x} + \color{red}{12} $$ |
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