Add $x^2-12x$ and $ \dfrac{35}{x^2} $ to get $ \dfrac{ \color{purple}{ x^4-12x^3+35 } }{ x^2 }$.

Step 1: Write $ x^2-12x $ 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 $\color{blue}{ x^2 }$.

$$ \begin{aligned} x^2-12x+ \frac{35}{x^2} & \xlongequal{\text{Step 1}} \frac{x^2-12x}{\color{red}{1}} + \frac{35}{x^2} = \frac{ \left( x^2-12x \right) \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} + \frac{ 35 }{ x^2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^4-12x^3 } }{ x^2 } + \frac{ \color{purple}{ 35 } }{ x^2 }=\frac{ \color{purple}{ x^4-12x^3+35 } }{ x^2 } \end{aligned} $$

Subtract $25$ from $ \dfrac{x^4-12x^3+35}{x^2} $ to get $ \dfrac{ \color{purple}{ x^4-12x^3-25x^2+35 } }{ x^2 }$.

Step 1: Write $ 25 $ 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 $\color{blue}{x^2}$.

$$ \begin{aligned} \frac{x^4-12x^3+35}{x^2} -25 & \xlongequal{\text{Step 1}} \frac{x^4-12x^3+35}{x^2} - \frac{25}{\color{red}{1}} = \frac{ x^4-12x^3+35 }{ x^2 } - \frac{ 25 \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^4-12x^3+35 } }{ x^2 } - \frac{ \color{purple}{ 25x^2 } }{ x^2 }=\frac{ \color{purple}{ x^4-12x^3-25x^2+35 } }{ x^2 } \end{aligned} $$