Answer
$$(h+3v)\cdot(1+h^2)+(v+3h)\cdot(1+v^2)=h^3+3h^2v+3hv^2+v^3+4h+4v$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(h+3v)\cdot(1+h^2)+(v+3h)\cdot(1+v^2)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}h+h^3+3v+3h^2v+v+v^3+3h+3hv^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}h^3+3h^2v+3hv^2+v^3+4h+4v\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{h+3v}\right) $ by each term in $ \left( 1+h^2\right) $. $$ \left( \color{blue}{h+3v}\right) \cdot \left( 1+h^2\right) = h+h^3+3v+3h^2v $$Multiply each term of $ \left( \color{blue}{v+3h}\right) $ by each term in $ \left( 1+v^2\right) $. $$ \left( \color{blue}{v+3h}\right) \cdot \left( 1+v^2\right) = v+v^3+3h+3hv^2 $$ |
| ② | Combine like terms: $$ \color{blue}{h} +h^3+ \color{red}{3v} +3h^2v+ \color{red}{v} +v^3+ \color{blue}{3h} +3hv^2 = h^3+3h^2v+3hv^2+v^3+ \color{blue}{4h} + \color{red}{4v} $$ |
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