Question
Answer
$$2n(8k+4)-k+1+(n-1)(14k+8)+(n-1)(n-2)(8k+4)=8kn^2+6kn+4n^2+k+4n+1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}2n(8k+4)-k+1+(n-1)(14k+8)+(n-1)(n-2)(8k+4)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}16kn+8n-k+1+14kn+8n-14k-8+(1n^2-2n-n+2)(8k+4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}16kn+8n-k+1+14kn+8n-14k-8+(1n^2-3n+2)(8k+4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}30kn-15k+16n-7+(1n^2-3n+2)(8k+4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}30kn-15k+16n-7+8kn^2+4n^2-24kn-12n+16k+8 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}8kn^2+6kn+4n^2+k+4n+1\end{aligned} $$ | |
| ① | Multiply $ \color{blue}{2n} $ by $ \left( 8k+4\right) $ $$ \color{blue}{2n} \cdot \left( 8k+4\right) = 16kn+8n $$ Multiply each term of $ \left( \color{blue}{n-1}\right) $ by each term in $ \left( 14k+8\right) $. $$ \left( \color{blue}{n-1}\right) \cdot \left( 14k+8\right) = 14kn+8n-14k-8 $$Multiply each term of $ \left( \color{blue}{n-1}\right) $ by each term in $ \left( n-2\right) $. $$ \left( \color{blue}{n-1}\right) \cdot \left( n-2\right) = n^2-2n-n+2 $$ |
| ② | Combine like terms: $$ n^2 \color{blue}{-2n} \color{blue}{-n} +2 = n^2 \color{blue}{-3n} +2 $$ |
| ③ | Combine like terms: $$ \color{blue}{16kn} + \color{red}{8n} \color{green}{-k} + \color{orange}{1} + \color{blue}{14kn} + \color{red}{8n} \color{green}{-14k} \color{orange}{-8} = \\ = \color{blue}{30kn} \color{green}{-15k} + \color{red}{16n} \color{orange}{-7} $$ |
| ④ | Multiply each term of $ \left( \color{blue}{n^2-3n+2}\right) $ by each term in $ \left( 8k+4\right) $. $$ \left( \color{blue}{n^2-3n+2}\right) \cdot \left( 8k+4\right) = 8kn^2+4n^2-24kn-12n+16k+8 $$ |
| ⑤ | Combine like terms: $$ \color{blue}{30kn} \color{red}{-15k} + \color{green}{16n} \color{orange}{-7} +8kn^2+4n^2 \color{blue}{-24kn} \color{green}{-12n} + \color{red}{16k} + \color{orange}{8} = \\ = 8kn^2+ \color{blue}{6kn} +4n^2+ \color{red}{k} + \color{green}{4n} + \color{orange}{1} $$ |
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