Answer
$$\frac{3-i}{2+i}+(1+i)\cdot(1-i)\cdot(3-4i)=-9i+7$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3-i}{2+i}+(1+i)\cdot(1-i)\cdot(3-4i)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}1-i+(1+i)\cdot(1-i)\cdot(3-4i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}1-i+(1-i+i-i^2)\cdot(3-4i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}1-i+(-i^2+1)\cdot(3-4i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}1-i+(1+1)\cdot(3-4i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}1-i+2\cdot(3-4i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}1-i+6-8i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}-9i+7\end{aligned} $$ | |
| ① | Divide $ \, 3-i \, $ by $ \, 2+i \, $ to get $\,\, 1-i $. ( view steps ) |
| ② | Multiply each term of $ \left( \color{blue}{1+i}\right) $ by each term in $ \left( 1-i\right) $. $$ \left( \color{blue}{1+i}\right) \cdot \left( 1-i\right) = 1 -\cancel{i}+ \cancel{i}-i^2 $$ |
| ③ | Combine like terms: $$ 1 \, \color{blue}{ -\cancel{i}} \,+ \, \color{blue}{ \cancel{i}} \,-i^2 = -i^2+1 $$ |
| ④ | $$ -i^2 = -(-1) = 1 $$ |
| ⑤ | Combine like terms: $$ \color{blue}{1} + \color{blue}{1} = \color{blue}{2} $$ |
| ⑥ | Multiply $ \color{blue}{2} $ by $ \left( 3-4i\right) $ $$ \color{blue}{2} \cdot \left( 3-4i\right) = 6-8i $$ |
| ⑦ | Combine like terms: $$ \color{blue}{1} \color{red}{-i} + \color{blue}{6} \color{red}{-8i} = \color{red}{-9i} + \color{blue}{7} $$ |
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