Find $ \left(-4+4i\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}{ 4 } $ and $ B = \color{red}{ 4i }$.

$$ \begin{aligned}\left(-4+4i\right)^2& \xlongequal{ S1 } \left(4-4i\right)^2 \xlongequal{ S2 } \color{blue}{4^2} -2 \cdot 4 \cdot 4i + \color{red}{\left( 4i \right)^2} = \\[1 em] & = 16-32i+16i^2\end{aligned} $$

$$ 16i^2 = 16 \cdot (-1) = -16 $$

Combine like terms:

$$ \, \color{blue}{ \cancel{16}} \,-32i \, \color{blue}{ -\cancel{16}} \, = -32i $$

Multiply $ \dfrac{1}{500} $ by $ 7+4i $ to get $ \dfrac{4i+7}{500} $.

Step 1: Write $ 7+4i $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{500} \cdot 7+4i & \xlongequal{\text{Step 1}} \frac{1}{500} \cdot \frac{7+4i}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot \left( 7+4i \right) }{ 500 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 7+4i }{ 500 } = \frac{4i+7}{500} \end{aligned} $$

Multiply $ \dfrac{4i+7}{500} $ by $ -32i $ to get $ \dfrac{ -128i^2-224i }{ 500 } $.

Step 1: Write $ -32i $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{4i+7}{500} \cdot -32i & \xlongequal{\text{Step 1}} \frac{4i+7}{500} \cdot \frac{-32i}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( 4i+7 \right) \cdot \left( -32i \right) }{ 500 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ -128i^2-224i }{ 500 } \end{aligned} $$

$$ -128i^2 = -128 \cdot (-1) = 128 $$

Multiply $ \dfrac{128-224i}{500} $ by $ 10+20i $ to get $ \dfrac{-4480i^2+320i+1280}{500} $.

Step 1: Write $ 10+20i $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{128-224i}{500} \cdot 10+20i & \xlongequal{\text{Step 1}} \frac{128-224i}{500} \cdot \frac{10+20i}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( 128-224i \right) \cdot \left( 10+20i \right) }{ 500 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 1280+2560i-2240i-4480i^2 }{ 500 } = \frac{-4480i^2+320i+1280}{500} \end{aligned} $$

$$ -4480i^2 = -4480 \cdot (-1) = 4480 $$

Simplify numerator

$$ \color{blue}{4480} +320i+ \color{blue}{1280} = 320i+ \color{blue}{5760} $$