Simplify $ \dfrac{x^2+6x+8}{x+4} $ to $ x+2$.

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

$$ \begin{aligned} \frac{x^2+6x+8}{x+4} & =\frac{ \left( x+2 \right) \cdot \color{blue}{ \left( x+4 \right) }}{ 1 \cdot \color{blue}{ \left( x+4 \right) }} = \\[1ex] &= \frac{x+2}{1} =x+2 \end{aligned} $$

Divide $ \dfrac{x^2+10x+16}{4x^2+36} $ by $ x+2 $ to get $ \dfrac{ x+8 }{ 4x^2+36 } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Factor numerators and denominators.

Step 3: Cancel common factors.

Step 4: Multiply numerators and denominators.

Step 5: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{x^2+10x+16}{4x^2+36} }{x+2} & \xlongequal{\text{Step 1}} \frac{x^2+10x+16}{4x^2+36} \cdot \frac{\color{blue}{1}}{\color{blue}{x+2}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x+8 \right) \cdot \color{blue}{ \left( x+2 \right) } }{ 4x^2+36 } \cdot \frac{ 1 }{ 1 \cdot \color{blue}{ \left( x+2 \right) } } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x+8 }{ 4x^2+36 } \cdot \frac{ 1 }{ 1 } \xlongequal{\text{Step 4}} \frac{ \left( x+8 \right) \cdot 1 }{ \left( 4x^2+36 \right) \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 5}} \frac{ x+8 }{ 4x^2+36 } \end{aligned} $$