Simplify (2x+14)/(2x2-98)

The numerator is $2x+14$. Factor out the common factor of 2:

$$2x+14 = 2(x+7)$$

The denominator is $2x^2-98$. First, factor out the common factor of 2:

$$2x^2-98 = 2(x^2-49)$$

Recognize that $x^2-49$ is a difference of squares, $a^2-b^2 = (a-b)(a+b)$, where $a=x$ and $b=7$:

$$x^2-49 = (x-7)(x+7)$$

So, the fully factored denominator is:

$$2(x-7)(x+7)$$

Both the numerator and the denominator have factors of 2 and $(x+7)$.

$$\frac{\cancel{2}\cancel{(x+7)}}{\cancel{2}(x-7)\cancel{(x+7)}} = \frac{1}{x-7}$$

The simplified expression is:

$$\frac{1}{x-7}$$