Subtract $ \dfrac{x}{6} $ from $ \dfrac{2}{x^2+14x+48} $ to get $ \dfrac{ \color{purple}{ -x^3-14x^2-48x+12 } }{ 6x^2+84x+288 }$.

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}{ 6 }$ and the second by $\color{blue}{ x^2+14x+48 }$.

$$ \begin{aligned} \frac{2}{x^2+14x+48} - \frac{x}{6} & = \frac{ 2 \cdot \color{blue}{ 6 }}{ \left( x^2+14x+48 \right) \cdot \color{blue}{ 6 }} - \frac{ x \cdot \color{blue}{ \left( x^2+14x+48 \right) }}{ 6 \cdot \color{blue}{ \left( x^2+14x+48 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 12 } }{ 6x^2+84x+288 } - \frac{ \color{purple}{ x^3+14x^2+48x } }{ 6x^2+84x+288 } = \\[1ex] &=\frac{ \color{purple}{ 12 - \left( x^3+14x^2+48x \right) } }{ 6x^2+84x+288 }=\frac{ \color{purple}{ -x^3-14x^2-48x+12 } }{ 6x^2+84x+288 } \end{aligned} $$