Find $ \left(2+i\right)^2 $ using 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}{ i }$.

$$ \begin{aligned}\left(2+i\right)^2 = \color{blue}{2^2} +2 \cdot 2 \cdot i + \color{red}{i^2} = 4+4i+i^2\end{aligned} $$

$$ i^2 = -1 $$

Combine like terms:

$$ \color{blue}{4} +4i \color{blue}{-1} = 4i+ \color{blue}{3} $$

Multiply $4i+3$ by $ \dfrac{3-i}{2} $ to get $ \dfrac{-4i^2+9i+9}{2} $.

Step 1: Write $ 4i+3 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 4i+3 \cdot \frac{3-i}{2} & \xlongequal{\text{Step 1}} \frac{4i+3}{\color{red}{1}} \cdot \frac{3-i}{2} \xlongequal{\text{Step 2}} \frac{ \left( 4i+3 \right) \cdot \left( 3-i \right) }{ 1 \cdot 2 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 12i-4i^2+9-3i }{ 2 } = \frac{-4i^2+9i+9}{2} \end{aligned} $$

$$ -4i^2 = -4 \cdot (-1) = 4 $$

Combine like terms:

$$ \color{blue}{4} +9i+ \color{blue}{9} = 9i+ \color{blue}{13} $$

Subtract $3i$ from $ \dfrac{9i+13}{2} $ to get $ \dfrac{ \color{purple}{ 3i+13 } }{ 2 }$.

Step 1: Write $ 3i $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{2}$.

$$ \begin{aligned} \frac{9i+13}{2} -3i & \xlongequal{\text{Step 1}} \frac{9i+13}{2} - \frac{3i}{\color{red}{1}} = \frac{ 9i+13 }{ 2 } - \frac{ 3i \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 9i+13 } }{ 2 } - \frac{ \color{purple}{ 6i } }{ 2 }=\frac{ \color{purple}{ 3i+13 } }{ 2 } \end{aligned} $$