Question
Answer
$$2(v^2-h^2)\cdot(1+v^2)+((v+h)^2+4)(v(h+3v)-2\cdot(1+v^2))=h^3v+h^2v^2+3hv^3+3v^4-4h^2+4v^2-8$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}2(v^2-h^2)\cdot(1+v^2)+((v+h)^2+4)(v(h+3v)-2\cdot(1+v^2))& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}2(v^2-h^2)\cdot(1+v^2)+(1v^2+2hv+h^2+4)(v(h+3v)-2\cdot(1+v^2)) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(2v^2-2h^2)\cdot(1+v^2)+(1v^2+2hv+h^2+4)(1hv+3v^2-(2+2v^2)) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}2v^2+2v^4-2h^2-2h^2v^2+(1v^2+2hv+h^2+4)(1hv+3v^2-(2+2v^2)) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}2v^2+2v^4-2h^2-2h^2v^2+(1v^2+2hv+h^2+4)(1hv+3v^2-2-2v^2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}2v^2+2v^4-2h^2-2h^2v^2+(1v^2+2hv+h^2+4)(1hv+v^2-2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}2v^2+2v^4-2h^2-2h^2v^2+h^3v+3h^2v^2+3hv^3+v^4-2h^2+2v^2-8 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}h^3v+h^2v^2+3hv^3+3v^4-4h^2+4v^2-8\end{aligned} $$ | |
| ① | Find $ \left(v+h\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ v } $ and $ B = \color{red}{ h }$. $$ \begin{aligned}\left(v+h\right)^2 = \color{blue}{v^2} +2 \cdot v \cdot h + \color{red}{h^2} = v^2+2hv+h^2\end{aligned} $$ |
| ② | Multiply $ \color{blue}{2} $ by $ \left( v^2-h^2\right) $ $$ \color{blue}{2} \cdot \left( v^2-h^2\right) = 2v^2-2h^2 $$Multiply $ \color{blue}{v} $ by $ \left( h+3v\right) $ $$ \color{blue}{v} \cdot \left( h+3v\right) = hv+3v^2 $$Multiply $ \color{blue}{2} $ by $ \left( 1+v^2\right) $ $$ \color{blue}{2} \cdot \left( 1+v^2\right) = 2+2v^2 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{2v^2-2h^2}\right) $ by each term in $ \left( 1+v^2\right) $. $$ \left( \color{blue}{2v^2-2h^2}\right) \cdot \left( 1+v^2\right) = 2v^2+2v^4-2h^2-2h^2v^2 $$ |
| ④ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 2+2v^2 \right) = -2-2v^2 $$ |
| ⑤ | Combine like terms: $$ hv+ \color{blue}{3v^2} -2 \color{blue}{-2v^2} = hv+ \color{blue}{v^2} -2 $$ |
| ⑥ | Multiply each term of $ \left( \color{blue}{v^2+2hv+h^2+4}\right) $ by each term in $ \left( hv+v^2-2\right) $. $$ \left( \color{blue}{v^2+2hv+h^2+4}\right) \cdot \left( hv+v^2-2\right) = \\ = hv^3+v^4-2v^2+2h^2v^2+2hv^3 -\cancel{4hv}+h^3v+h^2v^2-2h^2+ \cancel{4hv}+4v^2-8 $$ |
| ⑦ | Combine like terms: $$ \color{blue}{hv^3} +v^4 \color{red}{-2v^2} + \color{green}{2h^2v^2} + \color{blue}{2hv^3} \, \color{orange}{ -\cancel{4hv}} \,+h^3v+ \color{green}{h^2v^2} -2h^2+ \, \color{orange}{ \cancel{4hv}} \,+ \color{red}{4v^2} -8 = \\ = h^3v+ \color{green}{3h^2v^2} + \color{blue}{3hv^3} +v^4-2h^2+ \color{red}{2v^2} -8 $$ |
| ⑧ | Combine like terms: $$ \color{blue}{2v^2} + \color{red}{2v^4} \color{green}{-2h^2} \color{orange}{-2h^2v^2} +h^3v+ \color{orange}{3h^2v^2} +3hv^3+ \color{red}{v^4} \color{green}{-2h^2} + \color{blue}{2v^2} -8 = \\ = h^3v+ \color{orange}{h^2v^2} +3hv^3+ \color{red}{3v^4} \color{green}{-4h^2} + \color{blue}{4v^2} -8 $$ |
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