Multiply $ \dfrac{81}{145} $ by $ i $ to get $ \dfrac{ 81i }{ 145 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{81}{145} \cdot i & \xlongequal{\text{Step 1}} \frac{81}{145} \cdot \frac{i}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 81 \cdot i }{ 145 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 81i }{ 145 } \end{aligned} $$

Subtract $ \dfrac{81i}{145} $ from $ \dfrac{43}{145} $ to get $ \dfrac{-81i+43}{145} $.

To subtract expressions with the same denominators, we subtract the numerators and write the result over the common denominator.

$$ \begin{aligned} \frac{43}{145} - \frac{81i}{145} & = \frac{43}{\color{blue}{145}} - \frac{81i}{\color{blue}{145}} =\frac{ 43 - 81i }{ \color{blue}{ 145 }} = \\[1ex] &= \frac{-81i+43}{145} \end{aligned} $$

Multiply $53+56i$ by $ \dfrac{-81i+43}{145} $ to get $ \dfrac{-4536i^2-1885i+2279}{145} $.

Step 1: Write $ 53+56i $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 53+56i \cdot \frac{-81i+43}{145} & \xlongequal{\text{Step 1}} \frac{53+56i}{\color{red}{1}} \cdot \frac{-81i+43}{145} \xlongequal{\text{Step 2}} \frac{ \left( 53+56i \right) \cdot \left( -81i+43 \right) }{ 1 \cdot 145 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ -4293i+2279-4536i^2+2408i }{ 145 } = \frac{-4536i^2-1885i+2279}{145} \end{aligned} $$

$$ -4536i^2 = -4536 \cdot (-1) = 4536 $$

Simplify numerator

$$ \color{blue}{4536} -1885i+ \color{blue}{2279} = -1885i+ \color{blue}{6815} $$