Answer
$$(t^2-1)^2(4t^2-1)+(3t^2-1)^2-4t^6=0$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(t^2-1)^2(4t^2-1)+(3t^2-1)^2-4t^6& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(1t^4-2t^2+1)(4t^2-1)+9t^4-6t^2+1-4t^6 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}4t^6-t^4-8t^4+2t^2+4t^2-1+9t^4-6t^2+1-4t^6 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}4t^6-9t^4+6t^2-1+9t^4-6t^2+1-4t^6 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}4t^6-4t^6 \xlongequal{ } \\[1 em] & \xlongequal{ } \cancel{4t^6} -\cancel{4t^6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}0\end{aligned} $$ | |
| ① | Find $ \left(t^2-1\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ t^2 } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(t^2-1\right)^2 = \color{blue}{\left( t^2 \right)^2} -2 \cdot t^2 \cdot 1 + \color{red}{1^2} = t^4-2t^2+1\end{aligned} $$Find $ \left(3t^2-1\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 3t^2 } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(3t^2-1\right)^2 = \color{blue}{\left( 3t^2 \right)^2} -2 \cdot 3t^2 \cdot 1 + \color{red}{1^2} = 9t^4-6t^2+1\end{aligned} $$ |
| ② | Multiply each term of $ \left( \color{blue}{t^4-2t^2+1}\right) $ by each term in $ \left( 4t^2-1\right) $. $$ \left( \color{blue}{t^4-2t^2+1}\right) \cdot \left( 4t^2-1\right) = 4t^6-t^4-8t^4+2t^2+4t^2-1 $$ |
| ③ | Combine like terms: $$ 4t^6 \color{blue}{-t^4} \color{blue}{-8t^4} + \color{red}{2t^2} + \color{red}{4t^2} -1 = 4t^6 \color{blue}{-9t^4} + \color{red}{6t^2} -1 $$ |
| ④ | Combine like terms: $$ 4t^6 \, \color{blue}{ -\cancel{9t^4}} \,+ \, \color{green}{ \cancel{6t^2}} \, \, \color{blue}{ -\cancel{1}} \,+ \, \color{blue}{ \cancel{9t^4}} \, \, \color{green}{ -\cancel{6t^2}} \,+ \, \color{blue}{ \cancel{1}} \, = 4t^6 $$ |
| ⑤ | Combine like terms: $$ \, \color{blue}{ \cancel{4t^6}} \, \, \color{blue}{ -\cancel{4t^6}} \, = 0 $$ |
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