$$\dfrac{(-2+15i)^2+16\cdot(-2+15i)+289}{(-2+15i+42.6)(-2+15i+18)(-2+15i)^2}=\dfrac{-605916+2196i}{708215705}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}\frac{(-2+15i)^2+16\cdot(-2+15i)+289}{(-2+15i+42.6)(-2+15i+18)(-2+15i)^2}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{4-60i+225i^2-32+240i+289}{(225i^2+840i+640)(4-60i+225i^2)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}\frac{4-60i-225-32+240i+289}{(-225+840i+640)(4-60i-225)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} \htmlClass{explanationCircle explanationCircle11}{\textcircled {11}} } }}}\frac{-60i-221-32+240i+289}{(840i+415)(-60i-221)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle12}{\textcircled {12}} \htmlClass{explanationCircle explanationCircle13}{\textcircled {13}} \htmlClass{explanationCircle explanationCircle14}{\textcircled {14}} } }}}\frac{180i-253+289}{(840i+415)(-60i-221)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle15}{\textcircled {15}} \htmlClass{explanationCircle explanationCircle16}{\textcircled {16}} } }}}\frac{180i+36}{-50400i^2-185640i-24900i-91715} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle17}{\textcircled {17}} } }}}\frac{180i+36}{50400-185640i-24900i-91715} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle18}{\textcircled {18}} } }}}\frac{180i+36}{-210540i-41315} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle19}{\textcircled {19}} } }}}\frac{-605916+2196i}{708215705}\end{aligned} $$
①	Find $ \left(-2+15i\right)^2 $ in two steps. S1: Change all signs inside bracket. S2: Apply formula $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$ where $ A = \color{blue}{ 2 } $ and $ B = \color{red}{ 15i }$. $$ \begin{aligned}\left(-2+15i\right)^2& \xlongequal{ S1 } \left(2-15i\right)^2 \xlongequal{ S2 } \color{blue}{2^2} -2 \cdot 2 \cdot 15i + \color{red}{\left( 15i \right)^2} = \\[1 em] & = 4-60i+225i^2\end{aligned} $$
②	Multiply $ \color{blue}{16} $ by $ \left( -2+15i\right) $ $$ \color{blue}{16} \cdot \left( -2+15i\right) = -32+240i $$
③	Multiply each term of $ \left( \color{blue}{-2+15i+42}\right) $ by each term in $ \left( -2+15i+18\right) $. $$ \left( \color{blue}{-2+15i+42}\right) \cdot \left( -2+15i+18\right) = 4-30i-36-30i+225i^2+270i-84+630i+756 $$
④	Combine like terms: $$ \color{blue}{4} \color{red}{-30i} \color{green}{-36} \color{orange}{-30i} +225i^2+ \color{blue}{270i} \color{red}{-84} + \color{blue}{630i} + \color{red}{756} = 225i^2+ \color{blue}{840i} + \color{red}{640} $$
⑤	Find $ \left(-2+15i\right)^2 $ in two steps. S1: Change all signs inside bracket. S2: Apply formula $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$ where $ A = \color{blue}{ 2 } $ and $ B = \color{red}{ 15i }$. $$ \begin{aligned}\left(-2+15i\right)^2& \xlongequal{ S1 } \left(2-15i\right)^2 \xlongequal{ S2 } \color{blue}{2^2} -2 \cdot 2 \cdot 15i + \color{red}{\left( 15i \right)^2} = \\[1 em] & = 4-60i+225i^2\end{aligned} $$
⑥	$$ 225i^2 = 225 \cdot (-1) = -225 $$
⑦	$$ 225i^2 = 225 \cdot (-1) = -225 $$
⑧	$$ 225i^2 = 225 \cdot (-1) = -225 $$
⑨	Combine like terms: $$ \color{blue}{4} -60i \color{blue}{-225} = -60i \color{blue}{-221} $$
⑩	Combine like terms: $$ \color{blue}{-225} +840i+ \color{blue}{640} = 840i+ \color{blue}{415} $$
⑪	Combine like terms: $$ \color{blue}{4} -60i \color{blue}{-225} = -60i \color{blue}{-221} $$
⑫	Combine like terms: $$ \color{blue}{-60i} \color{red}{-221} \color{red}{-32} + \color{blue}{240i} = \color{blue}{180i} \color{red}{-253} $$
⑬	Combine like terms: $$ \color{blue}{-225} +840i+ \color{blue}{640} = 840i+ \color{blue}{415} $$
⑭	Combine like terms: $$ \color{blue}{4} -60i \color{blue}{-225} = -60i \color{blue}{-221} $$
⑮	Combine like terms: $$ 180i \color{blue}{-253} + \color{blue}{289} = 180i+ \color{blue}{36} $$
⑯	Multiply each term of $ \left( \color{blue}{840i+415}\right) $ by each term in $ \left( -60i-221\right) $. $$ \left( \color{blue}{840i+415}\right) \cdot \left( -60i-221\right) = -50400i^2-185640i-24900i-91715 $$
⑰	$$ -50400i^2 = -50400 \cdot (-1) = 50400 $$
⑱	Simplify denominator $$ \color{blue}{50400} \color{red}{-185640i} \color{red}{-24900i} \color{blue}{-91715} = \color{red}{-210540i} \color{blue}{-41315} $$
⑲	Divide $ \, 36+180i \, $ by $ \, -41315-210540i \, $ to get $\,\, \dfrac{-605916+2196i}{708215705} $. ( view steps )

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