Question
Answer
$$(m+1)^4(5m^2+6m+5)+(m-1)^4(11m^2+18m+11)=16m^6+64m^4+96m^3+64m^2+16$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(m+1)^4(5m^2+6m+5)+(m-1)^4(11m^2+18m+11)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}(1m^4+4m^3+6m^2+4m+1)(5m^2+6m+5)+(1m^4-4m^3+6m^2-4m+1)(11m^2+18m+11) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}5m^6+26m^5+59m^4+76m^3+59m^2+26m+5+11m^6-26m^5+5m^4+20m^3+5m^2-26m+11 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle11}{\textcircled {11}} } }}}16m^6+64m^4+96m^3+64m^2+16\end{aligned} $$ | |
| ① | $$ (m+1)^4 = (m+1)^2 \cdot (m+1)^2 $$ |
| ② | Find $ \left(m+1\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ m } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(m+1\right)^2 = \color{blue}{m^2} +2 \cdot m \cdot 1 + \color{red}{1^2} = m^2+2m+1\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{m^2+2m+1}\right) $ by each term in $ \left( m^2+2m+1\right) $. $$ \left( \color{blue}{m^2+2m+1}\right) \cdot \left( m^2+2m+1\right) = m^4+2m^3+m^2+2m^3+4m^2+2m+m^2+2m+1 $$ |
| ④ | Combine like terms: $$ m^4+ \color{blue}{2m^3} + \color{red}{m^2} + \color{blue}{2m^3} + \color{green}{4m^2} + \color{orange}{2m} + \color{green}{m^2} + \color{orange}{2m} +1 = m^4+ \color{blue}{4m^3} + \color{green}{6m^2} + \color{orange}{4m} +1 $$$$ (m-1)^4 = (m-1)^2 \cdot (m-1)^2 $$ |
| ⑤ | Find $ \left(m-1\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ m } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(m-1\right)^2 = \color{blue}{m^2} -2 \cdot m \cdot 1 + \color{red}{1^2} = m^2-2m+1\end{aligned} $$ |
| ⑥ | Multiply each term of $ \left( \color{blue}{m^2-2m+1}\right) $ by each term in $ \left( m^2-2m+1\right) $. $$ \left( \color{blue}{m^2-2m+1}\right) \cdot \left( m^2-2m+1\right) = m^4-2m^3+m^2-2m^3+4m^2-2m+m^2-2m+1 $$ |
| ⑦ | Combine like terms: $$ m^4 \color{blue}{-2m^3} + \color{red}{m^2} \color{blue}{-2m^3} + \color{green}{4m^2} \color{orange}{-2m} + \color{green}{m^2} \color{orange}{-2m} +1 = m^4 \color{blue}{-4m^3} + \color{green}{6m^2} \color{orange}{-4m} +1 $$ |
| ⑧ | Multiply each term of $ \left( \color{blue}{m^4+4m^3+6m^2+4m+1}\right) $ by each term in $ \left( 5m^2+6m+5\right) $. $$ \left( \color{blue}{m^4+4m^3+6m^2+4m+1}\right) \cdot \left( 5m^2+6m+5\right) = \\ = 5m^6+6m^5+5m^4+20m^5+24m^4+20m^3+30m^4+36m^3+30m^2+20m^3+24m^2+20m+5m^2+6m+5 $$ |
| ⑨ | Combine like terms: $$ 5m^6+ \color{blue}{6m^5} + \color{red}{5m^4} + \color{blue}{20m^5} + \color{green}{24m^4} + \color{orange}{20m^3} + \color{green}{30m^4} + \color{blue}{36m^3} + \color{red}{30m^2} + \color{blue}{20m^3} + \color{green}{24m^2} + \color{orange}{20m} + \color{green}{5m^2} + \color{orange}{6m} +5 = \\ = 5m^6+ \color{blue}{26m^5} + \color{green}{59m^4} + \color{blue}{76m^3} + \color{green}{59m^2} + \color{orange}{26m} +5 $$Multiply each term of $ \left( \color{blue}{m^4-4m^3+6m^2-4m+1}\right) $ by each term in $ \left( 11m^2+18m+11\right) $. $$ \left( \color{blue}{m^4-4m^3+6m^2-4m+1}\right) \cdot \left( 11m^2+18m+11\right) = \\ = 11m^6+18m^5+11m^4-44m^5-72m^4-44m^3+66m^4+108m^3+66m^2-44m^3-72m^2-44m+11m^2+18m+11 $$ |
| ⑩ | Combine like terms: $$ 11m^6+ \color{blue}{18m^5} + \color{red}{11m^4} \color{blue}{-44m^5} \color{green}{-72m^4} \color{orange}{-44m^3} + \color{green}{66m^4} + \color{blue}{108m^3} + \color{red}{66m^2} \color{blue}{-44m^3} \color{green}{-72m^2} \color{orange}{-44m} + \color{green}{11m^2} + \color{orange}{18m} +11 = \\ = 11m^6 \color{blue}{-26m^5} + \color{green}{5m^4} + \color{blue}{20m^3} + \color{green}{5m^2} \color{orange}{-26m} +11 $$ |
| ⑪ | Combine like terms: $$ \color{blue}{5m^6} + \, \color{red}{ \cancel{26m^5}} \,+ \color{orange}{59m^4} + \color{blue}{76m^3} + \color{red}{59m^2} + \, \color{green}{ \cancel{26m}} \,+ \color{blue}{5} + \color{blue}{11m^6} \, \color{red}{ -\cancel{26m^5}} \,+ \color{orange}{5m^4} + \color{blue}{20m^3} + \color{red}{5m^2} \, \color{green}{ -\cancel{26m}} \,+ \color{blue}{11} = \\ = \color{blue}{16m^6} + \color{orange}{64m^4} + \color{blue}{96m^3} + \color{red}{64m^2} + \color{blue}{16} $$ |
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