Question
Answer
$$(1+u^2(r^2+2r+4))^2=r^4u^4+4r^3u^4+12r^2u^4+16ru^4+2r^2u^2+16u^4+4ru^2+8u^2+1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(1+u^2(r^2+2r+4))^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(1+r^2u^2+2ru^2+4u^2)^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(1r^2u^2+2ru^2+4u^2+1)^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}r^4u^4+4r^3u^4+12r^2u^4+16ru^4+2r^2u^2+16u^4+4ru^2+8u^2+1\end{aligned} $$ | |
| ① | Multiply $ \color{blue}{u^2} $ by $ \left( r^2+2r+4\right) $ $$ \color{blue}{u^2} \cdot \left( r^2+2r+4\right) = r^2u^2+2ru^2+4u^2 $$ |
| ② | Combine like terms: $$ 1+r^2u^2+2ru^2+4u^2 = r^2u^2+2ru^2+4u^2+1 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{r^2u^2+2ru^2+4u^2+1}\right) $ by each term in $ \left( r^2u^2+2ru^2+4u^2+1\right) $. $$ \left( \color{blue}{r^2u^2+2ru^2+4u^2+1}\right) \cdot \left( r^2u^2+2ru^2+4u^2+1\right) = \\ = r^4u^4+2r^3u^4+4r^2u^4+r^2u^2+2r^3u^4+4r^2u^4+8ru^4+2ru^2+4r^2u^4+8ru^4+16u^4+4u^2+r^2u^2+2ru^2+4u^2+1 $$ |
| ④ | Combine like terms: $$ r^4u^4+ \color{blue}{2r^3u^4} + \color{red}{4r^2u^4} + \color{green}{r^2u^2} + \color{blue}{2r^3u^4} + \color{orange}{4r^2u^4} + \color{blue}{8ru^4} + \color{red}{2ru^2} + \color{orange}{4r^2u^4} + \color{blue}{8ru^4} +16u^4+ \color{green}{4u^2} + \color{green}{r^2u^2} + \color{red}{2ru^2} + \color{green}{4u^2} +1 = \\ = r^4u^4+ \color{blue}{4r^3u^4} + \color{orange}{12r^2u^4} + \color{blue}{16ru^4} + \color{green}{2r^2u^2} +16u^4+ \color{red}{4ru^2} + \color{green}{8u^2} +1 $$ |
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