Answer
$$8x(1-x^2)^2-6x\cdot(1-x^2)+(2x^2-1)(3x-4x^3)=-x$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}8x(1-x^2)^2-6x\cdot(1-x^2)+(2x^2-1)(3x-4x^3)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}8x(1-2x^2+x^4)-6x\cdot(1-x^2)+(2x^2-1)(3x-4x^3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}8x-16x^3+8x^5-(6x-6x^3)+6x^3-8x^5-3x+4x^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}8x-16x^3+8x^5-(6x-6x^3)-8x^5+10x^3-3x \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}8x-16x^3+8x^5-6x+6x^3-8x^5+10x^3-3x \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}8x^5-10x^3+2x-8x^5+10x^3-3x \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}-x\end{aligned} $$ | |
| ① | Find $ \left(1-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}{ 1 } $ and $ B = \color{red}{ x^2 }$. $$ \begin{aligned}\left(1-x^2\right)^2 = \color{blue}{1^2} -2 \cdot 1 \cdot x^2 + \color{red}{\left( x^2 \right)^2} = 1-2x^2+x^4\end{aligned} $$ |
| ② | Multiply $ \color{blue}{8x} $ by $ \left( 1-2x^2+x^4\right) $ $$ \color{blue}{8x} \cdot \left( 1-2x^2+x^4\right) = 8x-16x^3+8x^5 $$Multiply $ \color{blue}{6x} $ by $ \left( 1-x^2\right) $ $$ \color{blue}{6x} \cdot \left( 1-x^2\right) = 6x-6x^3 $$ Multiply each term of $ \left( \color{blue}{2x^2-1}\right) $ by each term in $ \left( 3x-4x^3\right) $. $$ \left( \color{blue}{2x^2-1}\right) \cdot \left( 3x-4x^3\right) = 6x^3-8x^5-3x+4x^3 $$ |
| ③ | Combine like terms: $$ \color{blue}{6x^3} -8x^5-3x+ \color{blue}{4x^3} = -8x^5+ \color{blue}{10x^3} -3x $$ |
| ④ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 6x-6x^3 \right) = -6x+6x^3 $$ |
| ⑤ | Combine like terms: $$ \color{blue}{8x} \color{red}{-16x^3} +8x^5 \color{blue}{-6x} + \color{red}{6x^3} = 8x^5 \color{red}{-10x^3} + \color{blue}{2x} $$ |
| ⑥ | Combine like terms: $$ \, \color{blue}{ \cancel{8x^5}} \, \, \color{green}{ -\cancel{10x^3}} \,+ \color{blue}{2x} \, \color{blue}{ -\cancel{8x^5}} \,+ \, \color{green}{ \cancel{10x^3}} \, \color{blue}{-3x} = \color{blue}{-x} $$ |
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