Multiply $3$ by $ \dfrac{x}{x^2+5x+6} $ to get $ \dfrac{ 3x }{ x^2+5x+6 } $.

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.

$$ \begin{aligned} 3 \cdot \frac{x}{x^2+5x+6} & \xlongequal{\text{Step 1}} \frac{3}{\color{red}{1}} \cdot \frac{x}{x^2+5x+6} \xlongequal{\text{Step 2}} \frac{ 3 \cdot x }{ 1 \cdot \left( x^2+5x+6 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 3x }{ x^2+5x+6 } \end{aligned} $$

Multiply $2$ by $ \dfrac{x}{x^2+8x-16} $ to get $ \dfrac{ 2x }{ x^2+8x-16 } $.

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.

$$ \begin{aligned} 2 \cdot \frac{x}{x^2+8x-16} & \xlongequal{\text{Step 1}} \frac{2}{\color{red}{1}} \cdot \frac{x}{x^2+8x-16} \xlongequal{\text{Step 2}} \frac{ 2 \cdot x }{ 1 \cdot \left( x^2+8x-16 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 2x }{ x^2+8x-16 } \end{aligned} $$

Subtract $ \dfrac{2x}{x^2+8x-16} $ from $ \dfrac{3x}{x^2+5x+6} $ to get $ \dfrac{ \color{purple}{ x^3+14x^2-60x } }{ x^4+13x^3+30x^2-32x-96 }$.

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^2+8x-16 }$ and the second by $\color{blue}{ x^2+5x+6 }$.

$$ \begin{aligned} \frac{3x}{x^2+5x+6} - \frac{2x}{x^2+8x-16} & = \frac{ 3x \cdot \color{blue}{ \left( x^2+8x-16 \right) }}{ \left( x^2+5x+6 \right) \cdot \color{blue}{ \left( x^2+8x-16 \right) }} - \frac{ 2x \cdot \color{blue}{ \left( x^2+5x+6 \right) }}{ \left( x^2+8x-16 \right) \cdot \color{blue}{ \left( x^2+5x+6 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 3x^3+24x^2-48x } }{ x^4+8x^3-16x^2+5x^3+40x^2-80x+6x^2+48x-96 } - \frac{ \color{purple}{ 2x^3+10x^2+12x } }{ x^4+8x^3-16x^2+5x^3+40x^2-80x+6x^2+48x-96 } = \\[1ex] &=\frac{ \color{purple}{ 3x^3+24x^2-48x - \left( 2x^3+10x^2+12x \right) } }{ x^4+13x^3+30x^2-32x-96 }=\frac{ \color{purple}{ x^3+14x^2-60x } }{ x^4+13x^3+30x^2-32x-96 } \end{aligned} $$