Simplify $ \dfrac{x^2+5x+6}{x^2+6x+9} $ to $ \dfrac{x+2}{x+3} $.

Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{x+3}$.

$$ \begin{aligned} \frac{x^2+5x+6}{x^2+6x+9} & =\frac{ \left( x+2 \right) \cdot \color{blue}{ \left( x+3 \right) }}{ \left( x+3 \right) \cdot \color{blue}{ \left( x+3 \right) }} = \\[1ex] &= \frac{x+2}{x+3} \end{aligned} $$

Simplify $ \dfrac{12x^2-29x-8}{4x^2-3x-1} $ to $ \dfrac{3x-8}{x-1} $.

Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{4x+1}$.

$$ \begin{aligned} \frac{12x^2-29x-8}{4x^2-3x-1} & =\frac{ \left( 3x-8 \right) \cdot \color{blue}{ \left( 4x+1 \right) }}{ \left( x-1 \right) \cdot \color{blue}{ \left( 4x+1 \right) }} = \\[1ex] &= \frac{3x-8}{x-1} \end{aligned} $$

Multiply $ \dfrac{x+2}{x+3} $ by $ \dfrac{3x-8}{x-1} $ to get $ \dfrac{3x^2-2x-16}{x^2+2x-3} $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{x+2}{x+3} \cdot \frac{3x-8}{x-1} & \xlongequal{\text{Step 1}} \frac{ \left( x+2 \right) \cdot \left( 3x-8 \right) }{ \left( x+3 \right) \cdot \left( x-1 \right) } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 3x^2-8x+6x-16 }{ x^2-x+3x-3 } = \frac{3x^2-2x-16}{x^2+2x-3} \end{aligned} $$

Add $ \dfrac{3x^2-2x-16}{x^2+2x-3} $ and $ \dfrac{8}{x-1} $ to get $ \dfrac{ \color{purple}{ 3x^2+6x+8 } }{ x^2+2x-3 }$.

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

$$ \begin{aligned} \frac{3x^2-2x-16}{x^2+2x-3} + \frac{8}{x-1} & = \frac{ 3x^2-2x-16 }{ x^2+2x-3 } + \frac{ 8 \cdot \color{blue}{ \left( x+3 \right) }}{ \left( x-1 \right) \cdot \color{blue}{ \left( x+3 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 3x^2-2x-16 } }{ x^2+2x-3 } + \frac{ \color{purple}{ 8x+24 } }{ x^2+2x-3 }=\frac{ \color{purple}{ 3x^2+6x+8 } }{ x^2+2x-3 } \end{aligned} $$