Question
Answer
$$3r^2+6r+4+4(1+r)^4=4r^4+16r^3+27r^2+22r+8$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}3r^2+6r+4+4(1+r)^4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}3r^2+6r+4+4(1r^4+4r^3+6r^2+4r+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}3r^2+6r+4+4r^4+16r^3+24r^2+16r+4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}4r^4+16r^3+27r^2+22r+8\end{aligned} $$ | |
| ① | $$ (1+r)^4 = (1+r)^2 \cdot (1+r)^2 $$ |
| ② | Find $ \left(1+r\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}{ r }$. $$ \begin{aligned}\left(1+r\right)^2 = \color{blue}{1^2} +2 \cdot 1 \cdot r + \color{red}{r^2} = 1+2r+r^2\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{1+2r+r^2}\right) $ by each term in $ \left( 1+2r+r^2\right) $. $$ \left( \color{blue}{1+2r+r^2}\right) \cdot \left( 1+2r+r^2\right) = 1+2r+r^2+2r+4r^2+2r^3+r^2+2r^3+r^4 $$ |
| ④ | Combine like terms: $$ 1+ \color{blue}{2r} + \color{red}{r^2} + \color{blue}{2r} + \color{green}{4r^2} + \color{orange}{2r^3} + \color{green}{r^2} + \color{orange}{2r^3} +r^4 = r^4+ \color{orange}{4r^3} + \color{green}{6r^2} + \color{blue}{4r} +1 $$ |
| ⑤ | Multiply $ \color{blue}{4} $ by $ \left( r^4+4r^3+6r^2+4r+1\right) $ $$ \color{blue}{4} \cdot \left( r^4+4r^3+6r^2+4r+1\right) = 4r^4+16r^3+24r^2+16r+4 $$ |
| ⑥ | Combine like terms: $$ \color{blue}{3r^2} + \color{red}{6r} + \color{green}{4} +4r^4+16r^3+ \color{blue}{24r^2} + \color{red}{16r} + \color{green}{4} = 4r^4+16r^3+ \color{blue}{27r^2} + \color{red}{22r} + \color{green}{8} $$ |
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