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x+1\geq 0 x-4\leq 0
For the product to be ≤0, one of the values x+1 and x-4 has to be ≥0 and the other has to be ≤0. Consider the case when x+1\geq 0 and x-4\leq 0.
x\in \begin{bmatrix}-1,4\end{bmatrix}
The solution satisfying both inequalities is x\in \left[-1,4\right].
x-4\geq 0 x+1\leq 0
Consider the case when x+1\leq 0 and x-4\geq 0.
x\in \emptyset
This is false for any x.
x\in \begin{bmatrix}-1,4\end{bmatrix}
The final solution is the union of the obtained solutions.