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