Subtract $ \dfrac{4}{x^2} $ from $ 8 $ to get $ \dfrac{ \color{purple}{ 8x^2-4 } }{ x^2 }$.

Step 1: Write $ 8 $ 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 first fraction by $\color{blue}{ x^2 }$.

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

Add $8x+7$ and $ \dfrac{4}{x} $ to get $ \dfrac{ \color{purple}{ 8x^2+7x+4 } }{ x }$.

Step 1: Write $ 8x+7 $ 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 $\color{blue}{ x }$.

$$ \begin{aligned} 8x+7+ \frac{4}{x} & \xlongequal{\text{Step 1}} \frac{8x+7}{\color{red}{1}} + \frac{4}{x} = \frac{ \left( 8x+7 \right) \cdot \color{blue}{ x }}{ 1 \cdot \color{blue}{ x }} + \frac{ 4 }{ x } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 8x^2+7x } }{ x } + \frac{ \color{purple}{ 4 } }{ x }=\frac{ \color{purple}{ 8x^2+7x+4 } }{ x } \end{aligned} $$

Multiply $9x^2+8$ by $ \dfrac{8x^2-4}{x^2} $ to get $ \dfrac{72x^4+28x^2-32}{x^2} $.

Step 1: Write $ 9x^2+8 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 9x^2+8 \cdot \frac{8x^2-4}{x^2} & \xlongequal{\text{Step 1}} \frac{9x^2+8}{\color{red}{1}} \cdot \frac{8x^2-4}{x^2} \xlongequal{\text{Step 2}} \frac{ \left( 9x^2+8 \right) \cdot \left( 8x^2-4 \right) }{ 1 \cdot x^2 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 72x^4-36x^2+64x^2-32 }{ x^2 } = \frac{72x^4+28x^2-32}{x^2} \end{aligned} $$

Multiply $ \dfrac{8x^2+7x+4}{x} $ by $ 18x $ to get $ \dfrac{ 144x^3+126x^2+72x }{ x } $.

Step 1: Write $ 18x $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{8x^2+7x+4}{x} \cdot 18x & \xlongequal{\text{Step 1}} \frac{8x^2+7x+4}{x} \cdot \frac{18x}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( 8x^2+7x+4 \right) \cdot 18x }{ x \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 144x^3+126x^2+72x }{ x } \end{aligned} $$

Add $ \dfrac{72x^4+28x^2-32}{x^2} $ and $ \dfrac{144x^3+126x^2+72x}{x} $ to get $ \dfrac{ \color{purple}{ 216x^5+126x^4+100x^3-32x } }{ x^3 }$.

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 }$ and the second by $\color{blue}{ x^2 }$.

$$ \begin{aligned} \frac{72x^4+28x^2-32}{x^2} + \frac{144x^3+126x^2+72x}{x} & = \frac{ \left( 72x^4+28x^2-32 \right) \cdot \color{blue}{ x }}{ x^2 \cdot \color{blue}{ x }} + \frac{ \left( 144x^3+126x^2+72x \right) \cdot \color{blue}{ x^2 }}{ x \cdot \color{blue}{ x^2 }} = \\[1ex] &=\frac{ \color{purple}{ 72x^5+28x^3-32x } }{ x^3 } + \frac{ \color{purple}{ 144x^5+126x^4+72x^3 } }{ x^3 } = \\[1ex] &=\frac{ \color{purple}{ 216x^5+126x^4+100x^3-32x } }{ x^3 } \end{aligned} $$