Answer
$$(1+i)\cdot(3-i)-(3+i)\cdot(1-i)=4i$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(1+i)\cdot(3-i)-(3+i)\cdot(1-i)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}3-i+3i-i^2-(3-3i+i-i^2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}-i^2+2i+3-(-i^2-2i+3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}1+2i+3-(1-2i+3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}2i+4-(-2i+4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}2i+4+2i-4 \xlongequal{ } \\[1 em] & \xlongequal{ }2i+ \cancel{4}+2i -\cancel{4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}4i\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{1+i}\right) $ by each term in $ \left( 3-i\right) $. $$ \left( \color{blue}{1+i}\right) \cdot \left( 3-i\right) = 3-i+3i-i^2 $$Multiply each term of $ \left( \color{blue}{3+i}\right) $ by each term in $ \left( 1-i\right) $. $$ \left( \color{blue}{3+i}\right) \cdot \left( 1-i\right) = 3-3i+i-i^2 $$ |
| ② | Combine like terms: $$ 3 \color{blue}{-i} + \color{blue}{3i} -i^2 = -i^2+ \color{blue}{2i} +3 $$Combine like terms: $$ 3 \color{blue}{-3i} + \color{blue}{i} -i^2 = -i^2 \color{blue}{-2i} +3 $$ |
| ③ | $$ -i^2 = -(-1) = 1 $$$$ -i^2 = -(-1) = 1 $$ |
| ④ | Combine like terms: $$ \color{blue}{1} +2i+ \color{blue}{3} = 2i+ \color{blue}{4} $$Combine like terms: $$ \color{blue}{1} -2i+ \color{blue}{3} = -2i+ \color{blue}{4} $$ |
| ⑤ | Remove the parentheses by changing the sign of each term within them. $$ - \left( -2i+4 \right) = 2i-4 $$ |
| ⑥ | Combine like terms: $$ \color{blue}{2i} + \, \color{red}{ \cancel{4}} \,+ \color{blue}{2i} \, \color{red}{ -\cancel{4}} \, = \color{blue}{4i} $$ |
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