Multiply $2$ by $ \dfrac{x}{5x+4} $ to get $ \dfrac{ 2x }{ 5x+4 } $.

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.

$$ \begin{aligned} 2 \cdot \frac{x}{5x+4} & \xlongequal{\text{Step 1}} \frac{2}{\color{red}{1}} \cdot \frac{x}{5x+4} \xlongequal{\text{Step 2}} \frac{ 2 \cdot x }{ 1 \cdot \left( 5x+4 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 2x }{ 5x+4 } \end{aligned} $$

Multiply $6$ by $ \dfrac{x}{2x+3} $ to get $ \dfrac{ 6x }{ 2x+3 } $.

Step 1: Write $ 6 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 6 \cdot \frac{x}{2x+3} & \xlongequal{\text{Step 1}} \frac{6}{\color{red}{1}} \cdot \frac{x}{2x+3} \xlongequal{\text{Step 2}} \frac{ 6 \cdot x }{ 1 \cdot \left( 2x+3 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 6x }{ 2x+3 } \end{aligned} $$

Add $ \dfrac{2x}{5x+4} $ and $ \dfrac{6x}{2x+3} $ to get $ \dfrac{ \color{purple}{ 34x^2+30x } }{ 10x^2+23x+12 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ 2x+3 }$ and the second by $\color{blue}{ 5x+4 }$.

$$ \begin{aligned} \frac{2x}{5x+4} + \frac{6x}{2x+3} & = \frac{ 2x \cdot \color{blue}{ \left( 2x+3 \right) }}{ \left( 5x+4 \right) \cdot \color{blue}{ \left( 2x+3 \right) }} + \frac{ 6x \cdot \color{blue}{ \left( 5x+4 \right) }}{ \left( 2x+3 \right) \cdot \color{blue}{ \left( 5x+4 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 4x^2+6x } }{ 10x^2+15x+8x+12 } + \frac{ \color{purple}{ 30x^2+24x } }{ 10x^2+15x+8x+12 } = \\[1ex] &=\frac{ \color{purple}{ 34x^2+30x } }{ 10x^2+23x+12 } \end{aligned} $$