Multiply $ \dfrac{2}{3} $ by $ i $ to get $ \dfrac{ 2i }{ 3 } $.

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.

$$ \begin{aligned} \frac{2}{3} \cdot i & \xlongequal{\text{Step 1}} \frac{2}{3} \cdot \frac{i}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 2 \cdot i }{ 3 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ 2i }{ 3 } \end{aligned} $$

Multiply $ \dfrac{2i}{3} $ by $ 4-5i $ to get $ \dfrac{-10i^2+8i}{3} $.

Step 1: Write $ 4-5i $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{2i}{3} \cdot 4-5i & \xlongequal{\text{Step 1}} \frac{2i}{3} \cdot \frac{4-5i}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 2i \cdot \left( 4-5i \right) }{ 3 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 8i-10i^2 }{ 3 } = \frac{-10i^2+8i}{3} \end{aligned} $$

$$ -10i^2 = -10 \cdot (-1) = 10 $$