Use the distributive property.

Add $ \dfrac{x^2}{x^2-81} $ and $ \dfrac{x}{x+9} $ to get $ \dfrac{ \color{purple}{ 2x^2-9x } }{ x^2-81 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{x-9}$.

$$ \begin{aligned} \frac{x^2}{x^2-81} + \frac{x}{x+9} & = \frac{ x^2 }{ x^2-81 } + \frac{ x \cdot \color{blue}{ \left( x-9 \right) }}{ \left( x+9 \right) \cdot \color{blue}{ \left( x-9 \right) }} = \\[1ex] &=\frac{ \color{purple}{ x^2 } }{ x^2-81 } + \frac{ \color{purple}{ x^2-9x } }{ x^2-81 }=\frac{ \color{purple}{ 2x^2-9x } }{ x^2-81 } \end{aligned} $$

Multiply $ \dfrac{2x^2-9x}{x^2-81} $ by $ 9 $ to get $ \dfrac{ 18x^2-81x }{ x^2-81 } $.

Step 1: Write $ 9 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{2x^2-9x}{x^2-81} \cdot 9 & \xlongequal{\text{Step 1}} \frac{2x^2-9x}{x^2-81} \cdot \frac{9}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( 2x^2-9x \right) \cdot 9 }{ \left( x^2-81 \right) \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 18x^2-81x }{ x^2-81 } \end{aligned} $$

Add $ \dfrac{18x^2-81x}{x^2-81} $ and $ \dfrac{9}{x-9} $ to get $ \dfrac{ \color{purple}{ 18x^2-72x+81 } }{ x^2-81 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{x+9}$.

$$ \begin{aligned} \frac{18x^2-81x}{x^2-81} + \frac{9}{x-9} & = \frac{ 18x^2-81x }{ x^2-81 } + \frac{ 9 \cdot \color{blue}{ \left( x+9 \right) }}{ \left( x-9 \right) \cdot \color{blue}{ \left( x+9 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 18x^2-81x } }{ x^2-81 } + \frac{ \color{purple}{ 9x+81 } }{ x^2-81 }=\frac{ \color{purple}{ 18x^2-72x+81 } }{ x^2-81 } \end{aligned} $$