Answer
$$2(2t+1)((t+1)(t+2)(2t+3)+15t+78)-3(t^2+6t+8)^2=5t^4+4t^3-26t^2+104t-24$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}2(2t+1)((t+1)(t+2)(2t+3)+15t+78)-3(t^2+6t+8)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}2(2t+1)((t+1)(t+2)(2t+3)+15t+78)-3(1t^4+12t^3+52t^2+96t+64) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(4t+2)((1t^2+2t+t+2)(2t+3)+15t+78)-(3t^4+36t^3+156t^2+288t+192) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(4t+2)((1t^2+3t+2)(2t+3)+15t+78)-(3t^4+36t^3+156t^2+288t+192) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}(4t+2)(2t^3+3t^2+6t^2+9t+4t+6+15t+78)-(3t^4+36t^3+156t^2+288t+192) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}(4t+2)(2t^3+9t^2+28t+84)-(3t^4+36t^3+156t^2+288t+192) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}8t^4+40t^3+130t^2+392t+168-(3t^4+36t^3+156t^2+288t+192) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}8t^4+40t^3+130t^2+392t+168-3t^4-36t^3-156t^2-288t-192 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}5t^4+4t^3-26t^2+104t-24\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{t^2+6t+8}\right) $ by each term in $ \left( t^2+6t+8\right) $. $$ \left( \color{blue}{t^2+6t+8}\right) \cdot \left( t^2+6t+8\right) = t^4+6t^3+8t^2+6t^3+36t^2+48t+8t^2+48t+64 $$ |
| ② | Combine like terms: $$ t^4+ \color{blue}{6t^3} + \color{red}{8t^2} + \color{blue}{6t^3} + \color{green}{36t^2} + \color{orange}{48t} + \color{green}{8t^2} + \color{orange}{48t} +64 = \\ = t^4+ \color{blue}{12t^3} + \color{green}{52t^2} + \color{orange}{96t} +64 $$ |
| ③ | Multiply $ \color{blue}{2} $ by $ \left( 2t+1\right) $ $$ \color{blue}{2} \cdot \left( 2t+1\right) = 4t+2 $$ Multiply each term of $ \left( \color{blue}{t+1}\right) $ by each term in $ \left( t+2\right) $. $$ \left( \color{blue}{t+1}\right) \cdot \left( t+2\right) = t^2+2t+t+2 $$Multiply $ \color{blue}{3} $ by $ \left( t^4+12t^3+52t^2+96t+64\right) $ $$ \color{blue}{3} \cdot \left( t^4+12t^3+52t^2+96t+64\right) = 3t^4+36t^3+156t^2+288t+192 $$ |
| ④ | Combine like terms: $$ t^2+ \color{blue}{2t} + \color{blue}{t} +2 = t^2+ \color{blue}{3t} +2 $$ |
| ⑤ | Multiply each term of $ \left( \color{blue}{t^2+3t+2}\right) $ by each term in $ \left( 2t+3\right) $. $$ \left( \color{blue}{t^2+3t+2}\right) \cdot \left( 2t+3\right) = 2t^3+3t^2+6t^2+9t+4t+6 $$ |
| ⑥ | Combine like terms: $$ 2t^3+ \color{blue}{3t^2} + \color{blue}{6t^2} + \color{red}{9t} + \color{green}{4t} + \color{orange}{6} + \color{green}{15t} + \color{orange}{78} = 2t^3+ \color{blue}{9t^2} + \color{green}{28t} + \color{orange}{84} $$ |
| ⑦ | Multiply each term of $ \left( \color{blue}{4t+2}\right) $ by each term in $ \left( 2t^3+9t^2+28t+84\right) $. $$ \left( \color{blue}{4t+2}\right) \cdot \left( 2t^3+9t^2+28t+84\right) = 8t^4+36t^3+112t^2+336t+4t^3+18t^2+56t+168 $$ |
| ⑧ | Combine like terms: $$ 8t^4+ \color{blue}{36t^3} + \color{red}{112t^2} + \color{green}{336t} + \color{blue}{4t^3} + \color{red}{18t^2} + \color{green}{56t} +168 = \\ = 8t^4+ \color{blue}{40t^3} + \color{red}{130t^2} + \color{green}{392t} +168 $$ |
| ⑨ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 3t^4+36t^3+156t^2+288t+192 \right) = -3t^4-36t^3-156t^2-288t-192 $$ |
| ⑩ | Combine like terms: $$ \color{blue}{8t^4} + \color{red}{40t^3} + \color{green}{130t^2} + \color{orange}{392t} + \color{blue}{168} \color{blue}{-3t^4} \color{red}{-36t^3} \color{green}{-156t^2} \color{orange}{-288t} \color{blue}{-192} = \\ = \color{blue}{5t^4} + \color{red}{4t^3} \color{green}{-26t^2} + \color{orange}{104t} \color{blue}{-24} $$ |
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