Find the vertical asymptote

Set the denominator = 0 and solve for $x+4=0$ ; we get $x=-4$ ; thus, the vertical asymptote is $x=-4$ .

Vertical asymptotes occur when

$\displaystyle \lim_{x \rightarrow c^+} f(x) = \pm \infty \quad\quad\quad\quad\quad \lim_{x \rightarrow c^-} f(x)= \pm \infty$

For rational functions, this occurs when the denominator is zero.

For the denominator to be zero, $x + 4 = 0 \implies x=-4$

Notice that:

$\displaystyle \lim_{x \rightarrow -4^+} \dfrac{1}{x+4}= \infty \quad$ and $\quad \displaystyle \lim_{x \rightarrow -4^-} \dfrac{1}{x+4}= - \infty$

This shows that a vertical asymptote occurs at $x=-4$

Therefore, $a=\boxed{-4}$