Find $ \left(2+x\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}{ x }$.

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

Add $8+2x$ and $ \dfrac{4}{x} $ to get $ \dfrac{ \color{purple}{ 2x^2+8x+4 } }{ x }$.

Step 1: Write $ 8+2x $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

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

$$ \begin{aligned} 8+2x+ \frac{4}{x} & \xlongequal{\text{Step 1}} \frac{8+2x}{\color{red}{1}} + \frac{4}{x} = \frac{ \left( 8+2x \right) \cdot \color{blue}{ x }}{ 1 \cdot \color{blue}{ x }} + \frac{ 4 }{ x } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 8x+2x^2 } }{ x } + \frac{ \color{purple}{ 4 } }{ x }=\frac{ \color{purple}{ 2x^2+8x+4 } }{ x } \end{aligned} $$

Multiply $4+4x+x^2$ by $ \dfrac{2x^2+8x+4}{x} $ to get $ \dfrac{2x^4+16x^3+44x^2+48x+16}{x} $.

Step 1: Write $ 4+4x+x^2 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 4+4x+x^2 \cdot \frac{2x^2+8x+4}{x} & \xlongequal{\text{Step 1}} \frac{4+4x+x^2}{\color{red}{1}} \cdot \frac{2x^2+8x+4}{x} \xlongequal{\text{Step 2}} \frac{ \left( 4+4x+x^2 \right) \cdot \left( 2x^2+8x+4 \right) }{ 1 \cdot x } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 8x^2+32x+16+8x^3+32x^2+16x+2x^4+8x^3+4x^2 }{ x } = \frac{2x^4+16x^3+44x^2+48x+16}{x} \end{aligned} $$