Multiply $30$ by $ \dfrac{x^3}{x^2} $ to get $ \dfrac{ 30x^3 }{ x^2 } $.

Step 1: Write $ 30 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 30 \cdot \frac{x^3}{x^2} & \xlongequal{\text{Step 1}} \frac{30}{\color{red}{1}} \cdot \frac{x^3}{x^2} \xlongequal{\text{Step 2}} \frac{ 30 \cdot x^3 }{ 1 \cdot x^2 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 30x^3 }{ x^2 } \end{aligned} $$

Divide both the top and bottom numbers by $ \color{orangered}{ 4 } $.

Subtract $ \dfrac{30x^3}{x^2} $ from $ x^5-7x^4 $ to get $ \dfrac{ \color{purple}{ x^7-7x^6-30x^3 } }{ x^2 }$.

Step 1: Write $ x^5-7x^4 $ 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 $\color{blue}{ x^2 }$.

$$ \begin{aligned} x^5-7x^4- \frac{30x^3}{x^2} & \xlongequal{\text{Step 1}} \frac{x^5-7x^4}{\color{red}{1}} - \frac{30x^3}{x^2} = \frac{ \left( x^5-7x^4 \right) \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} - \frac{ 30x^3 }{ x^2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^7-7x^6 } }{ x^2 } - \frac{ \color{purple}{ 30x^3 } }{ x^2 }=\frac{ \color{purple}{ x^7-7x^6-30x^3 } }{ x^2 } \end{aligned} $$

Add $ \dfrac{x^7-7x^6-30x^3}{x^2} $ and $ 10x $ to get $ \dfrac{ \color{purple}{ x^7-7x^6-20x^3 } }{ x^2 }$.

Step 1: Write $ 10x $ 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 $\color{blue}{x^2}$.

$$ \begin{aligned} \frac{x^7-7x^6-30x^3}{x^2} +10x & \xlongequal{\text{Step 1}} \frac{x^7-7x^6-30x^3}{x^2} + \frac{10x}{\color{red}{1}} = \frac{ x^7-7x^6-30x^3 }{ x^2 } + \frac{ 10x \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^7-7x^6-30x^3 } }{ x^2 } + \frac{ \color{purple}{ 10x^3 } }{ x^2 }=\frac{ \color{purple}{ x^7-7x^6-20x^3 } }{ x^2 } \end{aligned} $$

Remove 1 from denominator.

Add $ \dfrac{x^7-7x^6-20x^3}{x^2} $ and $ 214x^2 $ to get $ \dfrac{ \color{purple}{ x^7-7x^6+214x^4-20x^3 } }{ x^2 }$.

Step 1: Write $ 214x^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 second fraction by $\color{blue}{x^2}$.

$$ \begin{aligned} \frac{x^7-7x^6-20x^3}{x^2} +214x^2 & \xlongequal{\text{Step 1}} \frac{x^7-7x^6-20x^3}{x^2} + \frac{214x^2}{\color{red}{1}} = \frac{ x^7-7x^6-20x^3 }{ x^2 } + \frac{ 214x^2 \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^7-7x^6-20x^3 } }{ x^2 } + \frac{ \color{purple}{ 214x^4 } }{ x^2 }=\frac{ \color{purple}{ x^7-7x^6+214x^4-20x^3 } }{ x^2 } \end{aligned} $$

Remove 1 from denominator.

Subtract $49x^3$ from $ \dfrac{x^7-7x^6+214x^4-20x^3}{x^2} $ to get $ \dfrac{ \color{purple}{ x^7-7x^6-49x^5+214x^4-20x^3 } }{ x^2 }$.

Step 1: Write $ 49x^3 $ 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 $\color{blue}{x^2}$.

$$ \begin{aligned} \frac{x^7-7x^6+214x^4-20x^3}{x^2} -49x^3 & \xlongequal{\text{Step 1}} \frac{x^7-7x^6+214x^4-20x^3}{x^2} - \frac{49x^3}{\color{red}{1}} = \\[1ex] &= \frac{ x^7-7x^6+214x^4-20x^3 }{ x^2 } - \frac{ 49x^3 \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^7-7x^6+214x^4-20x^3 } }{ x^2 } - \frac{ \color{purple}{ 49x^5 } }{ x^2 }=\frac{ \color{purple}{ x^7-7x^6-49x^5+214x^4-20x^3 } }{ x^2 } \end{aligned} $$

Subtract $68x^2$ from $ \dfrac{x^7-7x^6-49x^5+214x^4-20x^3}{x^2} $ to get $ \dfrac{ \color{purple}{ x^7-7x^6-49x^5+146x^4-20x^3 } }{ x^2 }$.

Step 1: Write $ 68x^2 $ 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 $\color{blue}{x^2}$.

$$ \begin{aligned} \frac{x^7-7x^6-49x^5+214x^4-20x^3}{x^2} -68x^2 & \xlongequal{\text{Step 1}} \frac{x^7-7x^6-49x^5+214x^4-20x^3}{x^2} - \frac{68x^2}{\color{red}{1}} = \\[1ex] &= \frac{ x^7-7x^6-49x^5+214x^4-20x^3 }{ x^2 } - \frac{ 68x^2 \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^7-7x^6-49x^5+214x^4-20x^3 } }{ x^2 } - \frac{ \color{purple}{ 68x^4 } }{ x^2 } = \\[1ex] &=\frac{ \color{purple}{ x^7-7x^6-49x^5+146x^4-20x^3 } }{ x^2 } \end{aligned} $$

Add $ \dfrac{x^7-7x^6-49x^5+146x^4-20x^3}{x^2} $ and $ 280x $ to get $ \dfrac{ \color{purple}{ x^7-7x^6-49x^5+146x^4+260x^3 } }{ x^2 }$.

Step 1: Write $ 280x $ 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 $\color{blue}{x^2}$.

$$ \begin{aligned} \frac{x^7-7x^6-49x^5+146x^4-20x^3}{x^2} +280x & \xlongequal{\text{Step 1}} \frac{x^7-7x^6-49x^5+146x^4-20x^3}{x^2} + \frac{280x}{\color{red}{1}} = \\[1ex] &= \frac{ x^7-7x^6-49x^5+146x^4-20x^3 }{ x^2 } + \frac{ 280x \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^7-7x^6-49x^5+146x^4-20x^3 } }{ x^2 } + \frac{ \color{purple}{ 280x^3 } }{ x^2 } = \\[1ex] &=\frac{ \color{purple}{ x^7-7x^6-49x^5+146x^4+260x^3 } }{ x^2 } \end{aligned} $$