Multiply $35$ by $ \dfrac{x^9}{5} $ to get $ \dfrac{ 35x^9 }{ 5 } $.

Step 1: Write $ 35 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 35 \cdot \frac{x^9}{5} & \xlongequal{\text{Step 1}} \frac{35}{\color{red}{1}} \cdot \frac{x^9}{5} \xlongequal{\text{Step 2}} \frac{ 35 \cdot x^9 }{ 1 \cdot 5 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 35x^9 }{ 5 } \end{aligned} $$

Multiply $ \dfrac{35x^9}{5} $ by $ x^9 $ to get $ \dfrac{ 35x^{18} }{ 5 } $.

Step 1: Write $ x^9 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{35x^9}{5} \cdot x^9 & \xlongequal{\text{Step 1}} \frac{35x^9}{5} \cdot \frac{x^9}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 35x^9 \cdot x^9 }{ 5 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 35x^{18} }{ 5 } \end{aligned} $$

Subtract $ \dfrac{35x^{18}}{5} $ from $ 5x^{11}+10x^{10} $ to get $ \dfrac{ \color{purple}{ -35x^{18}+25x^{11}+50x^{10} } }{ 5 }$.

Step 1: Write $ 5x^{11}+10x^{10} $ 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 $\color{blue}{ 5 }$.

$$ \begin{aligned} 5x^{11}+10x^{10}- \frac{35x^{18}}{5} & \xlongequal{\text{Step 1}} \frac{5x^{11}+10x^{10}}{\color{red}{1}} - \frac{35x^{18}}{5} = \frac{ \left( 5x^{11}+10x^{10} \right) \cdot \color{blue}{ 5 }}{ 1 \cdot \color{blue}{ 5 }} - \frac{ 35x^{18} }{ 5 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 25x^{11}+50x^{10} } }{ 5 } - \frac{ \color{purple}{ 35x^{18} } }{ 5 }=\frac{ \color{purple}{ -35x^{18}+25x^{11}+50x^{10} } }{ 5 } \end{aligned} $$