Subtract $ \dfrac{9}{x^2} $ from $ 7 $ to get $ \dfrac{ \color{purple}{ 7x^2-9 } }{ x^2 }$.

Step 1: Write $ 7 $ 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 $\color{blue}{ x^2 }$.

$$ \begin{aligned} 7- \frac{9}{x^2} & \xlongequal{\text{Step 1}} \frac{7}{\color{red}{1}} - \frac{9}{x^2} = \frac{ 7 \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} - \frac{ 9 }{ x^2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 7x^2 } }{ x^2 } - \frac{ \color{purple}{ 9 } }{ x^2 }=\frac{ \color{purple}{ 7x^2-9 } }{ x^2 } \end{aligned} $$

Add $7x+2$ and $ \dfrac{9}{x} $ to get $ \dfrac{ \color{purple}{ 7x^2+2x+9 } }{ x }$.

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

$$ \begin{aligned} 7x+2+ \frac{9}{x} & \xlongequal{\text{Step 1}} \frac{7x+2}{\color{red}{1}} + \frac{9}{x} = \frac{ \left( 7x+2 \right) \cdot \color{blue}{ x }}{ 1 \cdot \color{blue}{ x }} + \frac{ 9 }{ x } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 7x^2+2x } }{ x } + \frac{ \color{purple}{ 9 } }{ x }=\frac{ \color{purple}{ 7x^2+2x+9 } }{ x } \end{aligned} $$

Multiply $2x^2+7$ by $ \dfrac{7x^2-9}{x^2} $ to get $ \dfrac{14x^4+31x^2-63}{x^2} $.

Step 1: Write $ 2x^2+7 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 2x^2+7 \cdot \frac{7x^2-9}{x^2} & \xlongequal{\text{Step 1}} \frac{2x^2+7}{\color{red}{1}} \cdot \frac{7x^2-9}{x^2} \xlongequal{\text{Step 2}} \frac{ \left( 2x^2+7 \right) \cdot \left( 7x^2-9 \right) }{ 1 \cdot x^2 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 14x^4-18x^2+49x^2-63 }{ x^2 } = \frac{14x^4+31x^2-63}{x^2} \end{aligned} $$

Multiply $ \dfrac{7x^2+2x+9}{x} $ by $ 4x $ to get $ \dfrac{ 28x^3+8x^2+36x }{ x } $.

Step 1: Write $ 4x $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{7x^2+2x+9}{x} \cdot 4x & \xlongequal{\text{Step 1}} \frac{7x^2+2x+9}{x} \cdot \frac{4x}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( 7x^2+2x+9 \right) \cdot 4x }{ x \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 28x^3+8x^2+36x }{ x } \end{aligned} $$

Add $ \dfrac{14x^4+31x^2-63}{x^2} $ and $ \dfrac{28x^3+8x^2+36x}{x} $ to get $ \dfrac{ \color{purple}{ 42x^5+8x^4+67x^3-63x } }{ 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 $\color{blue}{ x^2 }$.

$$ \begin{aligned} \frac{14x^4+31x^2-63}{x^2} + \frac{28x^3+8x^2+36x}{x} & = \frac{ \left( 14x^4+31x^2-63 \right) \cdot \color{blue}{ x }}{ x^2 \cdot \color{blue}{ x }} + \frac{ \left( 28x^3+8x^2+36x \right) \cdot \color{blue}{ x^2 }}{ x \cdot \color{blue}{ x^2 }} = \\[1ex] &=\frac{ \color{purple}{ 14x^5+31x^3-63x } }{ x^3 } + \frac{ \color{purple}{ 28x^5+8x^4+36x^3 } }{ x^3 } = \\[1ex] &=\frac{ \color{purple}{ 42x^5+8x^4+67x^3-63x } }{ x^3 } \end{aligned} $$