Find the range of the function
\[h(x) = \frac{2(x + 7)(x - 3)}{x + 7}.\]
If $x \neq -7,$ then we can cancel the factors of $x + 7$ to get
\[h(x) = 2(x - 3).\]If $x$ were allowed to be any real number, then $2(x - 3)$ could also be any real number.  However, the function is not defined for $x = -7,$ so the function cannot take on the value $2(-7 - 3) = -20.$

Therefore, the range of the function is $\boxed{(-\infty,-20) \cup (-20,\infty)}.$