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Question

Answer

$(1 - 3 i) \cdot (2 - i) \cdot (2 + i) \cdot (1 + 3 i) = - 45 i^{2} + 5$

Explanation

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$(1 - 3 i) \cdot (2 - i) \cdot (2 + i) \cdot (1 + 3 i) \overset{◯}{1} (2 - i - 6 i + 3 i^{2}) \cdot (2 + i) \cdot (1 + 3 i) \overset{◯}{2} (3 i^{2} - 7 i + 2) \cdot (2 + i) \cdot (1 + 3 i) \overset{◯}{3} (- 3 - 7 i + 2) \cdot (2 + i) \cdot (1 + 3 i) \overset{◯}{4} (- 7 i - 1) \cdot (2 + i) \cdot (1 + 3 i) \overset{◯}{5} (- 14 i - 7 i^{2} - 2 - i) \cdot (1 + 3 i) \overset{◯}{6} (- 7 i^{2} - 15 i - 2) \cdot (1 + 3 i) \overset{◯}{7} (7 - 15 i - 2) \cdot (1 + 3 i) \overset{◯}{8} (- 15 i + 5) \cdot (1 + 3 i) \overset{◯}{9} - 15 i - 45 i^{2} + 5 + 15 i - 15 i - 45 i^{2} + 5 + 15 i \overset{◯}{10} - 45 i^{2} + 5$
①	Multiply each term of $(1 - 3 i)$ by each term in $(2 - i)$ . $(1 - 3 i) \cdot (2 - i) = 2 - i - 6 i + 3 i^{2}$
②	Combine like terms: $2 - i - 6 i + 3 i^{2} = 3 i^{2} - 7 i + 2$
③	$3 i^{2} = 3 \cdot (- 1) = - 3$
④	Combine like terms: $- 3 - 7 i + 2 = - 7 i - 1$
⑤	Multiply each term of $(- 7 i - 1)$ by each term in $(2 + i)$ . $(- 7 i - 1) \cdot (2 + i) = - 14 i - 7 i^{2} - 2 - i$
⑥	Combine like terms: $- 14 i - 7 i^{2} - 2 - i = - 7 i^{2} - 15 i - 2$
⑦	$- 7 i^{2} = - 7 \cdot (- 1) = 7$
⑧	Combine like terms: $7 - 15 i - 2 = - 15 i + 5$
⑨	Multiply each term of $(- 15 i + 5)$ by each term in $(1 + 3 i)$ . $(- 15 i + 5) \cdot (1 + 3 i) = - 15 i - 45 i^{2} + 5 + 15 i$
⑩	Combine like terms: $- 15 i - 45 i^{2} + 5 + 15 i = - 45 i^{2} + 5$

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