Add $ \dfrac{x}{5x^2-45x+5} $ and $ 6 $ to get $ \dfrac{ \color{purple}{ 30x^2-269x+30 } }{ 5x^2-45x+5 }$.

Step 1: Write $ 6 $ 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 $\color{blue}{5x^2-45x+5}$.

$$ \begin{aligned} \frac{x}{5x^2-45x+5} +6 & \xlongequal{\text{Step 1}} \frac{x}{5x^2-45x+5} + \frac{6}{\color{red}{1}} = \frac{ x }{ 5x^2-45x+5 } + \frac{ 6 \cdot \color{blue}{ \left( 5x^2-45x+5 \right) }}{ 1 \cdot \color{blue}{ \left( 5x^2-45x+5 \right) }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x } }{ 5x^2-45x+5 } + \frac{ \color{purple}{ 30x^2-270x+30 } }{ 5x^2-45x+5 }=\frac{ \color{purple}{ 30x^2-269x+30 } }{ 5x^2-45x+5 } \end{aligned} $$