Subtract $x+5$ from $ \dfrac{x-2}{x+3} $ to get $ \dfrac{ \color{purple}{ -x^2-7x-17 } }{ x+3 }$.

Step 1: Write $ x+5 $ 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 $\color{blue}{x+3}$.

$$ \begin{aligned} \frac{x-2}{x+3} -x+5 & \xlongequal{\text{Step 1}} \frac{x-2}{x+3} - \frac{x+5}{\color{red}{1}} = \frac{ x-2 }{ x+3 } - \frac{ \left( x+5 \right) \cdot \color{blue}{ \left( x+3 \right) }}{ 1 \cdot \color{blue}{ \left( x+3 \right) }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x-2 } }{ x+3 } - \frac{ \color{purple}{ x^2+3x+5x+15 } }{ x+3 }=\frac{ \color{purple}{ x-2 - \left( x^2+3x+5x+15 \right) } }{ x+3 } = \\[1ex] &=\frac{ \color{purple}{ -x^2-7x-17 } }{ x+3 } \end{aligned} $$