Multiply $20$ by $ \dfrac{y+x}{y+v} $ to get $ \dfrac{20x+20y}{v+y} $.

Step 1: Write $ 20 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 20 \cdot \frac{y+x}{y+v} & \xlongequal{\text{Step 1}} \frac{20}{\color{red}{1}} \cdot \frac{y+x}{y+v} \xlongequal{\text{Step 2}} \frac{ 20 \cdot \left( y+x \right) }{ 1 \cdot \left( y+v \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 20y+20x }{ y+v } = \frac{20x+20y}{v+y} \end{aligned} $$

Multiply $ \color{blue}{5} $ by $ \left( 4y+v\right) $ $$ \color{blue}{5} \cdot \left( 4y+v\right) = 20y+5v $$

Multiply $ \dfrac{20x+20y}{v+y} $ by $ y $ to get $ \dfrac{ 20xy+20y^2 }{ v+y } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{20x+20y}{v+y} \cdot y & \xlongequal{\text{Step 1}} \frac{20x+20y}{v+y} \cdot \frac{y}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( 20x+20y \right) \cdot y }{ \left( v+y \right) \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 20xy+20y^2 }{ v+y } \end{aligned} $$

Multiply $ \color{blue}{5} $ by $ \left( 4y+v\right) $ $$ \color{blue}{5} \cdot \left( 4y+v\right) = 20y+5v $$

Subtract $ \dfrac{v}{20y+5v} $ from $ \dfrac{20xy+20y^2}{v+y} $ to get $ \dfrac{ \color{purple}{ 100vxy+100vy^2+400xy^2+400y^3-v^2-vy } }{ 5v^2+25vy+20y^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}{ 5v+20y }$ and the second by $\color{blue}{ v+y }$.

$$ \begin{aligned} \frac{20xy+20y^2}{v+y} - \frac{v}{20y+5v} & = \frac{ \left( 20xy+20y^2 \right) \cdot \color{blue}{ \left( 5v+20y \right) }}{ \left( v+y \right) \cdot \color{blue}{ \left( 5v+20y \right) }} - \frac{ v \cdot \color{blue}{ \left( v+y \right) }}{ \left( 20y+5v \right) \cdot \color{blue}{ \left( v+y \right) }} = \\[1ex] &=\frac{ \color{purple}{ 100vxy+400xy^2+100vy^2+400y^3 } }{ 5v^2+20vy+5vy+20y^2 } - \frac{ \color{purple}{ v^2+vy } }{ 5v^2+20vy+5vy+20y^2 } = \\[1ex] &=\frac{ \color{purple}{ 100vxy+400xy^2+100vy^2+400y^3 - \left( v^2+vy \right) } }{ 5v^2+25vy+20y^2 }=\frac{ \color{purple}{ 100vxy+100vy^2+400xy^2+400y^3-v^2-vy } }{ 5v^2+25vy+20y^2 } \end{aligned} $$