Subtract $ \dfrac{x}{x+1} $ from $ \dfrac{3}{2x+3} $ to get $ \dfrac{ \color{purple}{ -2x^2+3 } }{ 2x^2+5x+3 }$.

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}{ x+1 }$ and the second by $\color{blue}{ 2x+3 }$.

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