Answer
$(-4,-1)U(0,\infty)$
Work Step by Step
Step 1. Graph $R(x)=\frac{x+1}{x(x+4)}$ as shown in the figure.
Step 2. The vertical asymptotes ($x=-4,0$) and a zero ($x=-1$) divide the x-axis into four intervals $(-\infty,-4),(-4,-1),(-1,0),(0,\infty)$
Step 3. For $R(x)\gt0$, we can identify the solution intervals as $(-4,-1)U(0,\infty)$
