Answer
$$(t+1)(t+2)(2t+3)+15t+78=2t^3+9t^2+28t+84$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(t+1)(t+2)(2t+3)+15t+78& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(1t^2+2t+t+2)(2t+3)+15t+78 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(1t^2+3t+2)(2t+3)+15t+78 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}2t^3+3t^2+6t^2+9t+4t+6+15t+78 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}2t^3+9t^2+28t+84\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{t+1}\right) $ by each term in $ \left( t+2\right) $. $$ \left( \color{blue}{t+1}\right) \cdot \left( t+2\right) = t^2+2t+t+2 $$ |
| ② | Combine like terms: $$ t^2+ \color{blue}{2t} + \color{blue}{t} +2 = t^2+ \color{blue}{3t} +2 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{t^2+3t+2}\right) $ by each term in $ \left( 2t+3\right) $. $$ \left( \color{blue}{t^2+3t+2}\right) \cdot \left( 2t+3\right) = 2t^3+3t^2+6t^2+9t+4t+6 $$ |
| ④ | Combine like terms: $$ 2t^3+ \color{blue}{3t^2} + \color{blue}{6t^2} + \color{red}{9t} + \color{green}{4t} + \color{orange}{6} + \color{green}{15t} + \color{orange}{78} = 2t^3+ \color{blue}{9t^2} + \color{green}{28t} + \color{orange}{84} $$ |
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