Subtract $v+6$ from $ \dfrac{4}{5v-7} $ to get $ \dfrac{ \color{purple}{ -5v^2-23v+46 } }{ 5v-7 }$.

Step 1: Write $ v+6 $ 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 $\color{blue}{5v-7}$.

$$ \begin{aligned} \frac{4}{5v-7} -v+6 & \xlongequal{\text{Step 1}} \frac{4}{5v-7} - \frac{v+6}{\color{red}{1}} = \frac{ 4 }{ 5v-7 } - \frac{ \left( v+6 \right) \cdot \color{blue}{ \left( 5v-7 \right) }}{ 1 \cdot \color{blue}{ \left( 5v-7 \right) }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 4 } }{ 5v-7 } - \frac{ \color{purple}{ 5v^2-7v+30v-42 } }{ 5v-7 }=\frac{ \color{purple}{ 4 - \left( 5v^2-7v+30v-42 \right) } }{ 5v-7 } = \\[1ex] &=\frac{ \color{purple}{ -5v^2-23v+46 } }{ 5v-7 } \end{aligned} $$