Multiply $4$ by $ \dfrac{x}{6x^2+13x+6} $ to get $ \dfrac{ 4x }{ 6x^2+13x+6 } $.

Step 1: Write $ 4 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 4 \cdot \frac{x}{6x^2+13x+6} & \xlongequal{\text{Step 1}} \frac{4}{\color{red}{1}} \cdot \frac{x}{6x^2+13x+6} \xlongequal{\text{Step 2}} \frac{ 4 \cdot x }{ 1 \cdot \left( 6x^2+13x+6 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 4x }{ 6x^2+13x+6 } \end{aligned} $$

Multiply $3$ by $ \dfrac{x}{4x^2+9} $ to get $ \dfrac{ 3x }{ 4x^2+9 } $.

Step 1: Write $ 3 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 3 \cdot \frac{x}{4x^2+9} & \xlongequal{\text{Step 1}} \frac{3}{\color{red}{1}} \cdot \frac{x}{4x^2+9} \xlongequal{\text{Step 2}} \frac{ 3 \cdot x }{ 1 \cdot \left( 4x^2+9 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 3x }{ 4x^2+9 } \end{aligned} $$

Subtract $ \dfrac{3x}{4x^2+9} $ from $ \dfrac{4x}{6x^2+13x+6} $ to get $ \dfrac{ \color{purple}{ -2x^3-39x^2+18x } }{ 24x^4+52x^3+78x^2+117x+54 }$.

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}{ 4x^2+9 }$ and the second by $\color{blue}{ 6x^2+13x+6 }$.

$$ \begin{aligned} \frac{4x}{6x^2+13x+6} - \frac{3x}{4x^2+9} & = \frac{ 4x \cdot \color{blue}{ \left( 4x^2+9 \right) }}{ \left( 6x^2+13x+6 \right) \cdot \color{blue}{ \left( 4x^2+9 \right) }} - \frac{ 3x \cdot \color{blue}{ \left( 6x^2+13x+6 \right) }}{ \left( 4x^2+9 \right) \cdot \color{blue}{ \left( 6x^2+13x+6 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 16x^3+36x } }{ 24x^4+54x^2+52x^3+117x+24x^2+54 } - \frac{ \color{purple}{ 18x^3+39x^2+18x } }{ 24x^4+54x^2+52x^3+117x+24x^2+54 } = \\[1ex] &=\frac{ \color{purple}{ 16x^3+36x - \left( 18x^3+39x^2+18x \right) } }{ 24x^4+52x^3+78x^2+117x+54 }=\frac{ \color{purple}{ -2x^3-39x^2+18x } }{ 24x^4+52x^3+78x^2+117x+54 } \end{aligned} $$