Add $ \dfrac{3x-5}{4x^2+12x+9} $ and $ \dfrac{4}{2x+3} $ to get $ \dfrac{ \color{purple}{ 11x+7 } }{ 4x^2+12x+9 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{2x+3}$.

$$ \begin{aligned} \frac{3x-5}{4x^2+12x+9} + \frac{4}{2x+3} & = \frac{ 3x-5 }{ 4x^2+12x+9 } + \frac{ 4 \cdot \color{blue}{ \left( 2x+3 \right) }}{ \left( 2x+3 \right) \cdot \color{blue}{ \left( 2x+3 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 3x-5 } }{ 4x^2+12x+9 } + \frac{ \color{purple}{ 8x+12 } }{ 4x^2+12x+9 }=\frac{ \color{purple}{ 11x+7 } }{ 4x^2+12x+9 } \end{aligned} $$

Multiply $2$ by $ \dfrac{x}{3} $ to get $ \dfrac{ 2x }{ 3 } $.

Step 1: Write $ 2 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 2 \cdot \frac{x}{3} & \xlongequal{\text{Step 1}} \frac{2}{\color{red}{1}} \cdot \frac{x}{3} \xlongequal{\text{Step 2}} \frac{ 2 \cdot x }{ 1 \cdot 3 } \xlongequal{\text{Step 3}} \frac{ 2x }{ 3 } \end{aligned} $$

Subtract $ \dfrac{2x}{3} $ from $ \dfrac{11x+7}{4x^2+12x+9} $ to get $ \dfrac{ \color{purple}{ -8x^3-24x^2+15x+21 } }{ 12x^2+36x+27 }$.

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}{ 3 }$ and the second by $\color{blue}{ 4x^2+12x+9 }$.

$$ \begin{aligned} \frac{11x+7}{4x^2+12x+9} - \frac{2x}{3} & = \frac{ \left( 11x+7 \right) \cdot \color{blue}{ 3 }}{ \left( 4x^2+12x+9 \right) \cdot \color{blue}{ 3 }} - \frac{ 2x \cdot \color{blue}{ \left( 4x^2+12x+9 \right) }}{ 3 \cdot \color{blue}{ \left( 4x^2+12x+9 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 33x+21 } }{ 12x^2+36x+27 } - \frac{ \color{purple}{ 8x^3+24x^2+18x } }{ 12x^2+36x+27 } = \\[1ex] &=\frac{ \color{purple}{ 33x+21 - \left( 8x^3+24x^2+18x \right) } }{ 12x^2+36x+27 }=\frac{ \color{purple}{ -8x^3-24x^2+15x+21 } }{ 12x^2+36x+27 } \end{aligned} $$