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

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

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

Multiply $ \dfrac{3x^2+38x+24}{4x^2+45x+36} $ by $ \dfrac{x+6}{3x+2} $ to get $ \dfrac{x^2+18x+72}{4x^2+45x+36} $.

Step 1: Factor numerators and denominators.

Step 2: Cancel common factors.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} \frac{3x^2+38x+24}{4x^2+45x+36} \cdot \frac{x+6}{3x+2} & \xlongequal{\text{Step 1}} \frac{ \left( x+12 \right) \cdot \color{blue}{ \left( 3x+2 \right) } }{ 4x^2+45x+36 } \cdot \frac{ x+6 }{ 1 \cdot \color{blue}{ \left( 3x+2 \right) } } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ x+12 }{ 4x^2+45x+36 } \cdot \frac{ x+6 }{ 1 } \xlongequal{\text{Step 3}} \frac{ \left( x+12 \right) \cdot \left( x+6 \right) }{ \left( 4x^2+45x+36 \right) \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ x^2+6x+12x+72 }{ 4x^2+45x+36 } = \frac{x^2+18x+72}{4x^2+45x+36} \end{aligned} $$

Add $ \dfrac{x^2+18x+72}{4x^2+45x+36} $ and $ 4 $ to get $ \dfrac{ \color{purple}{ 17x^2+198x+216 } }{ 4x^2+45x+36 }$.

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

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{4x^2+45x+36}$.

$$ \begin{aligned} \frac{x^2+18x+72}{4x^2+45x+36} +4 & \xlongequal{\text{Step 1}} \frac{x^2+18x+72}{4x^2+45x+36} + \frac{4}{\color{red}{1}} = \frac{ x^2+18x+72 }{ 4x^2+45x+36 } + \frac{ 4 \cdot \color{blue}{ \left( 4x^2+45x+36 \right) }}{ 1 \cdot \color{blue}{ \left( 4x^2+45x+36 \right) }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^2+18x+72 } }{ 4x^2+45x+36 } + \frac{ \color{purple}{ 16x^2+180x+144 } }{ 4x^2+45x+36 } = \\[1ex] &=\frac{ \color{purple}{ 17x^2+198x+216 } }{ 4x^2+45x+36 } \end{aligned} $$