Add $7+x$ and $ \dfrac{5}{x} $ to get $ \dfrac{ \color{purple}{ x^2+7x+5 } }{ x }$.

Step 1: Write $ 7+x $ 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 $\color{blue}{ x }$.

$$ \begin{aligned} 7+x+ \frac{5}{x} & \xlongequal{\text{Step 1}} \frac{7+x}{\color{red}{1}} + \frac{5}{x} = \frac{ \left( 7+x \right) \cdot \color{blue}{ x }}{ 1 \cdot \color{blue}{ x }} + \frac{ 5 }{ x } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 7x+x^2 } }{ x } + \frac{ \color{purple}{ 5 } }{ x }=\frac{ \color{purple}{ x^2+7x+5 } }{ x } \end{aligned} $$

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 $ \dfrac{x^2+7x+5}{x} $ by $ 4+2x $ to get $ \dfrac{2x^3+18x^2+38x+20}{x} $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{x^2+7x+5}{x} \cdot 4+2x & \xlongequal{\text{Step 1}} \frac{x^2+7x+5}{x} \cdot \frac{4+2x}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( x^2+7x+5 \right) \cdot \left( 4+2x \right) }{ x \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 4x^2+2x^3+28x+14x^2+20+10x }{ x } = \frac{2x^3+18x^2+38x+20}{x} \end{aligned} $$

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 $ \dfrac{2x^3+18x^2+38x+20}{x} $ by $ \dfrac{2x^2+8x+4}{x} $ to get $ \dfrac{4x^5+52x^4+228x^3+416x^2+312x+80}{x^2} $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{2x^3+18x^2+38x+20}{x} \cdot \frac{2x^2+8x+4}{x} & \xlongequal{\text{Step 1}} \frac{ \left( 2x^3+18x^2+38x+20 \right) \cdot \left( 2x^2+8x+4 \right) }{ x \cdot x } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 4x^5+16x^4+8x^3+36x^4+144x^3+72x^2+76x^3+304x^2+152x+40x^2+160x+80 }{ x^2 } = \frac{4x^5+52x^4+228x^3+416x^2+312x+80}{x^2} \end{aligned} $$