Question
Answer
$$((2x(1-y-5))^2+20)^2=16x^4y^4+256x^4y^3+1536x^4y^2+4096x^4y+4096x^4+160x^2y^2+1280x^2y+2560x^2+400$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}((2x(1-y-5))^2+20)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}((2x(-y-4))^2+20)^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}((-2xy-8x)^2+20)^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(4x^2y^2+32x^2y+64x^2+20)^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}16x^4y^4+256x^4y^3+1536x^4y^2+4096x^4y+4096x^4+160x^2y^2+1280x^2y+2560x^2+400\end{aligned} $$ | |
| ① | Combine like terms: $$ \color{blue}{1} -y \color{blue}{-5} = -y \color{blue}{-4} $$ |
| ② | Multiply $ \color{blue}{2x} $ by $ \left( -y-4\right) $ $$ \color{blue}{2x} \cdot \left( -y-4\right) = -2xy-8x $$ |
| ③ | Find $ \left(-2xy-8x\right)^2 $ in two steps. S1: Change all signs inside bracket. S2: Apply formula $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 2xy } $ and $ B = \color{red}{ 8x }$. $$ \begin{aligned}\left(-2xy-8x\right)^2& \xlongequal{ S1 } \left(2xy+8x\right)^2 \xlongequal{ S2 } \color{blue}{\left( 2xy \right)^2} +2 \cdot 2xy \cdot 8x + \color{red}{\left( 8x \right)^2} = \\[1 em] & = 4x^2y^2+32x^2y+64x^2\end{aligned} $$ |
| ④ | Multiply each term of $ \left( \color{blue}{4x^2y^2+32x^2y+64x^2+20}\right) $ by each term in $ \left( 4x^2y^2+32x^2y+64x^2+20\right) $. $$ \left( \color{blue}{4x^2y^2+32x^2y+64x^2+20}\right) \cdot \left( 4x^2y^2+32x^2y+64x^2+20\right) = \\ = 16x^4y^4+128x^4y^3+256x^4y^2+80x^2y^2+128x^4y^3+1024x^4y^2+2048x^4y+640x^2y+256x^4y^2+2048x^4y+4096x^4+1280x^2+80x^2y^2+640x^2y+1280x^2+400 $$ |
| ⑤ | Combine like terms: $$ 16x^4y^4+ \color{blue}{128x^4y^3} + \color{red}{256x^4y^2} + \color{green}{80x^2y^2} + \color{blue}{128x^4y^3} + \color{orange}{1024x^4y^2} + \color{blue}{2048x^4y} + \color{red}{640x^2y} + \color{orange}{256x^4y^2} + \color{blue}{2048x^4y} +4096x^4+ \color{green}{1280x^2} + \color{green}{80x^2y^2} + \color{red}{640x^2y} + \color{green}{1280x^2} +400 = \\ = 16x^4y^4+ \color{blue}{256x^4y^3} + \color{orange}{1536x^4y^2} + \color{blue}{4096x^4y} +4096x^4+ \color{green}{160x^2y^2} + \color{red}{1280x^2y} + \color{green}{2560x^2} +400 $$ |
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