Multiply $ \color{blue}{19x} $ by $ \left( x+6\right) $ $$ \color{blue}{19x} \cdot \left( x+6\right) = 19x^2+114x $$

Multiply $3$ by $ \dfrac{x}{x^2} $ to get $ \dfrac{ 3x }{ x^2 } $.

Step 1: Write $ 3 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 3 \cdot \frac{x}{x^2} & \xlongequal{\text{Step 1}} \frac{3}{\color{red}{1}} \cdot \frac{x}{x^2} \xlongequal{\text{Step 2}} \frac{ 3 \cdot x }{ 1 \cdot x^2 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 3x }{ x^2 } \end{aligned} $$

Add $ \dfrac{3x}{x^2} $ and $ 9x $ to get $ \dfrac{ \color{purple}{ 9x^3+3x } }{ x^2 }$.

Step 1: Write $ 9x $ 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 $\color{blue}{x^2}$.

$$ \begin{aligned} \frac{3x}{x^2} +9x & \xlongequal{\text{Step 1}} \frac{3x}{x^2} + \frac{9x}{\color{red}{1}} = \frac{ 3x }{ x^2 } + \frac{ 9x \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 3x } }{ x^2 } + \frac{ \color{purple}{ 9x^3 } }{ x^2 }=\frac{ \color{purple}{ 9x^3+3x } }{ x^2 } \end{aligned} $$

Add $ \dfrac{9x^3+3x}{x^2} $ and $ 19x^2+114x $ to get $ \dfrac{ \color{purple}{ 19x^4+123x^3+3x } }{ x^2 }$.

Step 1: Write $ 19x^2+114x $ 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 $\color{blue}{x^2}$.

$$ \begin{aligned} \frac{9x^3+3x}{x^2} +19x^2+114x & \xlongequal{\text{Step 1}} \frac{9x^3+3x}{x^2} + \frac{19x^2+114x}{\color{red}{1}} = \frac{ 9x^3+3x }{ x^2 } + \frac{ \left( 19x^2+114x \right) \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 9x^3+3x } }{ x^2 } + \frac{ \color{purple}{ 19x^4+114x^3 } }{ x^2 }=\frac{ \color{purple}{ 19x^4+123x^3+3x } }{ x^2 } \end{aligned} $$