Multiply $6$ by $ \dfrac{x}{x^2-36} $ to get $ \dfrac{ 6x }{ x^2-36 } $.

Step 1: Write $ 6 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 6 \cdot \frac{x}{x^2-36} & \xlongequal{\text{Step 1}} \frac{6}{\color{red}{1}} \cdot \frac{x}{x^2-36} \xlongequal{\text{Step 2}} \frac{ 6 \cdot x }{ 1 \cdot \left( x^2-36 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 6x }{ x^2-36 } \end{aligned} $$

Add $ \dfrac{6x}{x^2-36} $ and $ \dfrac{x-6}{x+6} $ to get $ \dfrac{ \color{purple}{ x^2-6x+36 } }{ x^2-36 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{x-6}$.

$$ \begin{aligned} \frac{6x}{x^2-36} + \frac{x-6}{x+6} & = \frac{ 6x }{ x^2-36 } + \frac{ \left( x-6 \right) \cdot \color{blue}{ \left( x-6 \right) }}{ \left( x+6 \right) \cdot \color{blue}{ \left( x-6 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 6x } }{ x^2-36 } + \frac{ \color{purple}{ x^2-6x-6x+36 } }{ x^2-36 }=\frac{ \color{purple}{ x^2-6x+36 } }{ x^2-36 } \end{aligned} $$