We can start by rearranging the equation 5 sin x + 4 cos x = 0 by dividing both sides by cos(x):
5
sin
�
cos
�
+
4
=
0
5
cosx
sinx
+4=0
Recall that $\frac{\sin x}{\cos x} = \tan x$. So we can substitute this in:
5
tan
�
+
4
=
0
5tanx+4=0
Now we can solve for $\tan x$:
\begin{align*}
5 \tan x + 4 &= 0 \
5 \tan x &= -4 \
\tan x &= \frac{-4}{5}
\end{align*}
Therefore, the value of $\tan x$ is $\boxed{\frac{-4}{5}}$.