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Question

Answer

$\frac{1 - 4 i - ( 2 + 4 i ) \cdot ( - 3 - i )}{4 - i} = \frac{2 + 43 i}{17}$

Explanation

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$\frac{1 - 4 i - ( 2 + 4 i ) \cdot ( - 3 - i )}{4 - i} \overset{◯}{1} \frac{1 - 4 i - ( - 6 - 2 i - 12 i - 4 i ^{2} )}{4 - i} \overset{◯}{2} \frac{1 - 4 i - ( - 4 i ^{2} - 14 i - 6 )}{4 - i} \overset{◯}{3} \frac{1 - 4 i - ( 4 - 14 i - 6 )}{4 - i} \overset{◯}{4} \frac{1 - 4 i - ( - 14 i - 2 )}{4 - i} \overset{◯}{5} \frac{1 - 4 i + 14 i + 2}{4 - i} \overset{◯}{6} \frac{10 i + 3}{4 - i} \overset{◯}{7} \frac{2 + 43 i}{17}$
①	Multiply each term of $(2 + 4 i)$ by each term in $(- 3 - i)$ . $(2 + 4 i) \cdot (- 3 - i) = - 6 - 2 i - 12 i - 4 i^{2}$
②	Combine like terms: $- 6 - 2 i - 12 i - 4 i^{2} = - 4 i^{2} - 14 i - 6$
③	$- 4 i^{2} = - 4 \cdot (- 1) = 4$
④	Combine like terms: $4 - 14 i - 6 = - 14 i - 2$
⑤	Remove the parentheses by changing the sign of each term within them. $- (- 14 i - 2) = 14 i + 2$
⑥	Simplify numerator $1 - 4 i + 14 i + 2 = 10 i + 3$
⑦	Divide $3 + 10 i$ by $4 - i$ to get $\frac{2 + 43 i}{17}$ . ( view steps )

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