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+7x+4}{x} $ to get $ \dfrac{72x^4+63x^3+100x^2+56x+32}{x} $.

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.

$$ \begin{aligned} 9x^2+8 \cdot \frac{8x^2+7x+4}{x} & \xlongequal{\text{Step 1}} \frac{9x^2+8}{\color{red}{1}} \cdot \frac{8x^2+7x+4}{x} \xlongequal{\text{Step 2}} \frac{ \left( 9x^2+8 \right) \cdot \left( 8x^2+7x+4 \right) }{ 1 \cdot x } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 72x^4+63x^3+36x^2+64x^2+56x+32 }{ x } = \frac{72x^4+63x^3+100x^2+56x+32}{x} \end{aligned} $$