Subtract $ \dfrac{4}{n+1} $ from $ \dfrac{5}{5n-6} $ to get $ \dfrac{ \color{purple}{ -15n+29 } }{ 5n^2-n-6 }$.

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}{ n+1 }$ and the second by $\color{blue}{ 5n-6 }$.

$$ \begin{aligned} \frac{5}{5n-6} - \frac{4}{n+1} & = \frac{ 5 \cdot \color{blue}{ \left( n+1 \right) }}{ \left( 5n-6 \right) \cdot \color{blue}{ \left( n+1 \right) }} - \frac{ 4 \cdot \color{blue}{ \left( 5n-6 \right) }}{ \left( n+1 \right) \cdot \color{blue}{ \left( 5n-6 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 5n+5 } }{ 5n^2+5n-6n-6 } - \frac{ \color{purple}{ 20n-24 } }{ 5n^2+5n-6n-6 }=\frac{ \color{purple}{ 5n+5 - \left( 20n-24 \right) } }{ 5n^2-n-6 } = \\[1ex] &=\frac{ \color{purple}{ -15n+29 } }{ 5n^2-n-6 } \end{aligned} $$