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

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $x^3$ and $ \dfrac{2x^2}{x^2} $ to get $ \dfrac{ \color{purple}{ x^5+2x^2 } }{ x^2 }$.

Step 1: Write $ x^3 $ 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 $\color{blue}{ x^2 }$.

$$ \begin{aligned} x^3+ \frac{2x^2}{x^2} & \xlongequal{\text{Step 1}} \frac{x^3}{\color{red}{1}} + \frac{2x^2}{x^2} = \frac{ x^3 \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} + \frac{ 2x^2 }{ x^2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^5 } }{ x^2 } + \frac{ \color{purple}{ 2x^2 } }{ x^2 }=\frac{ \color{purple}{ x^5+2x^2 } }{ x^2 } \end{aligned} $$

Subtract $12x$ from $ \dfrac{x^5+2x^2}{x^2} $ to get $ \dfrac{ \color{purple}{ x^5-12x^3+2x^2 } }{ x^2 }$.

Step 1: Write $ 12x $ 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 second fraction by $\color{blue}{x^2}$.

$$ \begin{aligned} \frac{x^5+2x^2}{x^2} -12x & \xlongequal{\text{Step 1}} \frac{x^5+2x^2}{x^2} - \frac{12x}{\color{red}{1}} = \frac{ x^5+2x^2 }{ x^2 } - \frac{ 12x \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^5+2x^2 } }{ x^2 } - \frac{ \color{purple}{ 12x^3 } }{ x^2 }=\frac{ \color{purple}{ x^5-12x^3+2x^2 } }{ x^2 } \end{aligned} $$

Add $ \dfrac{x^5-12x^3+2x^2}{x^2} $ and $ 27 $ to get $ \dfrac{ \color{purple}{ x^5-12x^3+29x^2 } }{ x^2 }$.

Step 1: Write $ 27 $ 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 $\color{blue}{x^2}$.

$$ \begin{aligned} \frac{x^5-12x^3+2x^2}{x^2} +27 & \xlongequal{\text{Step 1}} \frac{x^5-12x^3+2x^2}{x^2} + \frac{27}{\color{red}{1}} = \frac{ x^5-12x^3+2x^2 }{ x^2 } + \frac{ 27 \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^5-12x^3+2x^2 } }{ x^2 } + \frac{ \color{purple}{ 27x^2 } }{ x^2 }=\frac{ \color{purple}{ x^5-12x^3+29x^2 } }{ x^2 } \end{aligned} $$