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Question

Answer

$(1 + 2 i) \cdot (3 + 2 i) + (5 - 2 i)^{2} = - 12 i + 20$

Explanation

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$(1 + 2 i) \cdot (3 + 2 i) + (5 - 2 i)^{2} \overset{◯}{1} (1 + 2 i) \cdot (3 + 2 i) + 25 - 20 i + 4 i^{2} \overset{◯}{2} (1 + 2 i) \cdot (3 + 2 i) + 25 - 20 i - 4 \overset{◯}{3} (1 + 2 i) \cdot (3 + 2 i) - 20 i + 21 \overset{◯}{4} 3 + 2 i + 6 i + 4 i^{2} - 20 i + 21 \overset{◯}{5} 4 i^{2} + 8 i + 3 - 20 i + 21 \overset{◯}{6} - 4 + 8 i + 3 - 20 i + 21 \overset{◯}{7} 8 i - 1 - 20 i + 21 \overset{◯}{8} - 12 i + 20$
①	Find $(5 - 2 i)^{2}$ using formula. $(A - B)^{2} = A^{2} - 2 \cdot A \cdot B + B^{2}$ where $A = 5$ and $B = 2 i$ . $(5 - 2 i)^{2} = 5^{2} - 2 \cdot 5 \cdot 2 i + (2 i)^{2} = 25 - 20 i + 4 i^{2}$
②	$4 i^{2} = 4 \cdot (- 1) = - 4$
③	Combine like terms: $25 - 20 i - 4 = - 20 i + 21$
④	Multiply each term of $(1 + 2 i)$ by each term in $(3 + 2 i)$ . $(1 + 2 i) \cdot (3 + 2 i) = 3 + 2 i + 6 i + 4 i^{2}$
⑤	Combine like terms: $3 + 2 i + 6 i + 4 i^{2} = 4 i^{2} + 8 i + 3$
⑥	$4 i^{2} = 4 \cdot (- 1) = - 4$
⑦	Combine like terms: $- 4 + 8 i + 3 = 8 i - 1$
⑧	Combine like terms: $8 i - 1 - 20 i + 21 = - 12 i + 20$

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