Subtract $ \dfrac{x+14}{x+7} $ from $ \dfrac{x+6}{x-2} $ to get $ \dfrac{ \color{purple}{ x+70 } }{ x^2+5x-14 }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ x+7 }$ and the second by $\color{blue}{ x-2 }$.

$$ \begin{aligned} \frac{x+6}{x-2} - \frac{x+14}{x+7} & = \frac{ \left( x+6 \right) \cdot \color{blue}{ \left( x+7 \right) }}{ \left( x-2 \right) \cdot \color{blue}{ \left( x+7 \right) }} - \frac{ \left( x+14 \right) \cdot \color{blue}{ \left( x-2 \right) }}{ \left( x+7 \right) \cdot \color{blue}{ \left( x-2 \right) }} = \\[1ex] &=\frac{ \color{purple}{ x^2+7x+6x+42 } }{ x^2+7x-2x-14 } - \frac{ \color{purple}{ x^2-2x+14x-28 } }{ x^2+7x-2x-14 } = \\[1ex] &=\frac{ \color{purple}{ x^2+7x+6x+42 - \left( x^2-2x+14x-28 \right) } }{ x^2+5x-14 }=\frac{ \color{purple}{ x+70 } }{ x^2+5x-14 } \end{aligned} $$

Divide $ \dfrac{x+70}{x^2+5x-14} $ by $ x+70 $ to get $ \dfrac{ 1 }{ x^2+5x-14 } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Cancel $ \color{blue}{ x+70 } $ in first and second fraction.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{x+70}{x^2+5x-14} }{x+70} & \xlongequal{\text{Step 1}} \frac{x+70}{x^2+5x-14} \cdot \frac{\color{blue}{1}}{\color{blue}{x+70}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{\color{blue}{1}}{x^2+5x-14} \cdot \frac{1}{\color{blue}{1}} \xlongequal{\text{Step 3}} \frac{ 1 \cdot 1 }{ \left( x^2+5x-14 \right) \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ 1 }{ x^2+5x-14 } \end{aligned} $$