Multiply $8$ by $ \dfrac{x}{x^2} $ to get $ \dfrac{ 8x }{ x^2 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 8 \cdot \frac{x}{x^2} & \xlongequal{\text{Step 1}} \frac{8}{\color{red}{1}} \cdot \frac{x}{x^2} \xlongequal{\text{Step 2}} \frac{ 8 \cdot x }{ 1 \cdot x^2 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 8x }{ x^2 } \end{aligned} $$

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

Step 1: Write $ x^2 $ 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 $\color{blue}{ x^2 }$.

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

Add $ \dfrac{x^4+8x}{x^2} $ and $ 8x $ to get $ \dfrac{ \color{purple}{ x^4+8x^3+8x } }{ x^2 }$.

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

$$ \begin{aligned} \frac{x^4+8x}{x^2} +8x & \xlongequal{\text{Step 1}} \frac{x^4+8x}{x^2} + \frac{8x}{\color{red}{1}} = \frac{ x^4+8x }{ x^2 } + \frac{ 8x \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^4+8x } }{ x^2 } + \frac{ \color{purple}{ 8x^3 } }{ x^2 }=\frac{ \color{purple}{ x^4+8x^3+8x } }{ x^2 } \end{aligned} $$

Add $ \dfrac{x^4+8x^3+8x}{x^2} $ and $ 7 $ to get $ \dfrac{ \color{purple}{ x^4+8x^3+7x^2+8x } }{ x^2 }$.

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

$$ \begin{aligned} \frac{x^4+8x^3+8x}{x^2} +7 & \xlongequal{\text{Step 1}} \frac{x^4+8x^3+8x}{x^2} + \frac{7}{\color{red}{1}} = \frac{ x^4+8x^3+8x }{ x^2 } + \frac{ 7 \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^4+8x^3+8x } }{ x^2 } + \frac{ \color{purple}{ 7x^2 } }{ x^2 }=\frac{ \color{purple}{ x^4+8x^3+7x^2+8x } }{ x^2 } \end{aligned} $$