Answer
$$(2n+3)^3-(2n+1)^3=24n^2+48n+26$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(2n+3)^3-(2n+1)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}8n^3+36n^2+54n+27-(8n^3+12n^2+6n+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}8n^3+36n^2+54n+27-8n^3-12n^2-6n-1 \xlongequal{ } \\[1 em] & \xlongequal{ } \cancel{8n^3}+36n^2+54n+27 -\cancel{8n^3}-12n^2-6n-1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}24n^2+48n+26\end{aligned} $$ | |
| ① | Find $ \left(2n+3\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = 2n $ and $ B = 3 $. $$ \left(2n+3\right)^3 = \left( 2n \right)^3+3 \cdot \left( 2n \right)^2 \cdot 3 + 3 \cdot 2n \cdot 3^2+3^3 = 8n^3+36n^2+54n+27 $$Find $ \left(2n+1\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = 2n $ and $ B = 1 $. $$ \left(2n+1\right)^3 = \left( 2n \right)^3+3 \cdot \left( 2n \right)^2 \cdot 1 + 3 \cdot 2n \cdot 1^2+1^3 = 8n^3+12n^2+6n+1 $$ |
| ② | Remove the parentheses by changing the sign of each term within them. $$ - \left( 8n^3+12n^2+6n+1 \right) = -8n^3-12n^2-6n-1 $$ |
| ③ | Combine like terms: $$ \, \color{blue}{ \cancel{8n^3}} \,+ \color{green}{36n^2} + \color{orange}{54n} + \color{blue}{27} \, \color{blue}{ -\cancel{8n^3}} \, \color{green}{-12n^2} \color{orange}{-6n} \color{blue}{-1} = \color{green}{24n^2} + \color{orange}{48n} + \color{blue}{26} $$ |
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