Question
Answer
$$4(r^2+2r+2)^2+(r+4)^2=4r^4+16r^3+33r^2+40r+32$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}4(r^2+2r+2)^2+(r+4)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}4(1r^4+4r^3+8r^2+8r+4)+r^2+8r+16 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}4r^4+16r^3+32r^2+32r+16+r^2+8r+16 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}4r^4+16r^3+33r^2+40r+32\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{r^2+2r+2}\right) $ by each term in $ \left( r^2+2r+2\right) $. $$ \left( \color{blue}{r^2+2r+2}\right) \cdot \left( r^2+2r+2\right) = r^4+2r^3+2r^2+2r^3+4r^2+4r+2r^2+4r+4 $$ |
| ② | Combine like terms: $$ r^4+ \color{blue}{2r^3} + \color{red}{2r^2} + \color{blue}{2r^3} + \color{green}{4r^2} + \color{orange}{4r} + \color{green}{2r^2} + \color{orange}{4r} +4 = r^4+ \color{blue}{4r^3} + \color{green}{8r^2} + \color{orange}{8r} +4 $$Find $ \left(r+4\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ r } $ and $ B = \color{red}{ 4 }$. $$ \begin{aligned}\left(r+4\right)^2 = \color{blue}{r^2} +2 \cdot r \cdot 4 + \color{red}{4^2} = r^2+8r+16\end{aligned} $$ |
| ③ | Multiply $ \color{blue}{4} $ by $ \left( r^4+4r^3+8r^2+8r+4\right) $ $$ \color{blue}{4} \cdot \left( r^4+4r^3+8r^2+8r+4\right) = 4r^4+16r^3+32r^2+32r+16 $$ |
| ④ | Combine like terms: $$ 4r^4+16r^3+ \color{blue}{32r^2} + \color{red}{32r} + \color{green}{16} + \color{blue}{r^2} + \color{red}{8r} + \color{green}{16} = \\ = 4r^4+16r^3+ \color{blue}{33r^2} + \color{red}{40r} + \color{green}{32} $$ |
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