Answer
Your friend is wrong because the vertical asymptote is $x=-5$, not $x=5$.
Work Step by Step
The function
$$f(x)=\dfrac{1}{x+5}+2$$
is defined for all real values of $x$ except for the denominator's zero(s).
$$x+5=0\Rightarrow x=-5.$$
The graph of $f$ has a vertical asymptote $x=-5$, not $x=5$. The friend is wrong.
