Answer
$$(1-3i)\cdot(2-i)\cdot(2+i)\cdot(1+3i)=-45i^2+5$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(1-3i)\cdot(2-i)\cdot(2+i)\cdot(1+3i)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(2-i-6i+3i^2)\cdot(2+i)\cdot(1+3i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(3i^2-7i+2)\cdot(2+i)\cdot(1+3i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(-3-7i+2)\cdot(2+i)\cdot(1+3i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(-7i-1)\cdot(2+i)\cdot(1+3i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}(-14i-7i^2-2-i)\cdot(1+3i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}(-7i^2-15i-2)\cdot(1+3i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}(7-15i-2)\cdot(1+3i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}(-15i+5)\cdot(1+3i) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}-15i-45i^2+5+15i \xlongequal{ } \\[1 em] & \xlongequal{ } -\cancel{15i}-45i^2+5+ \cancel{15i} \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}-45i^2+5\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{1-3i}\right) $ by each term in $ \left( 2-i\right) $. $$ \left( \color{blue}{1-3i}\right) \cdot \left( 2-i\right) = 2-i-6i+3i^2 $$ |
| ② | Combine like terms: $$ 2 \color{blue}{-i} \color{blue}{-6i} +3i^2 = 3i^2 \color{blue}{-7i} +2 $$ |
| ③ | $$ 3i^2 = 3 \cdot (-1) = -3 $$ |
| ④ | Combine like terms: $$ \color{blue}{-3} -7i+ \color{blue}{2} = -7i \color{blue}{-1} $$ |
| ⑤ | Multiply each term of $ \left( \color{blue}{-7i-1}\right) $ by each term in $ \left( 2+i\right) $. $$ \left( \color{blue}{-7i-1}\right) \cdot \left( 2+i\right) = -14i-7i^2-2-i $$ |
| ⑥ | Combine like terms: $$ \color{blue}{-14i} -7i^2-2 \color{blue}{-i} = -7i^2 \color{blue}{-15i} -2 $$ |
| ⑦ | $$ -7i^2 = -7 \cdot (-1) = 7 $$ |
| ⑧ | Combine like terms: $$ \color{blue}{7} -15i \color{blue}{-2} = -15i+ \color{blue}{5} $$ |
| ⑨ | Multiply each term of $ \left( \color{blue}{-15i+5}\right) $ by each term in $ \left( 1+3i\right) $. $$ \left( \color{blue}{-15i+5}\right) \cdot \left( 1+3i\right) = -\cancel{15i}-45i^2+5+ \cancel{15i} $$ |
| ⑩ | Combine like terms: $$ \, \color{blue}{ -\cancel{15i}} \,-45i^2+5+ \, \color{blue}{ \cancel{15i}} \, = -45i^2+5 $$ |
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