Add $10x+1$ and $ \dfrac{x}{7x^2-6x+7} $ to get $ \dfrac{ \color{purple}{ 70x^3-53x^2+65x+7 } }{ 7x^2-6x+7 }$.

Step 1: Write $ 10x+1 $ 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 first fraction by $\color{blue}{ 7x^2-6x+7 }$.

$$ \begin{aligned} 10x+1+ \frac{x}{7x^2-6x+7} & \xlongequal{\text{Step 1}} \frac{10x+1}{\color{red}{1}} + \frac{x}{7x^2-6x+7} = \frac{ \left( 10x+1 \right) \cdot \color{blue}{ \left( 7x^2-6x+7 \right) }}{ 1 \cdot \color{blue}{ \left( 7x^2-6x+7 \right) }} + \frac{ x }{ 7x^2-6x+7 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 70x^3-60x^2+70x+7x^2-6x+7 } }{ 7x^2-6x+7 } + \frac{ \color{purple}{ x } }{ 7x^2-6x+7 } = \\[1ex] &=\frac{ \color{purple}{ 70x^3-53x^2+65x+7 } }{ 7x^2-6x+7 } \end{aligned} $$