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-2x^{2}-10x\geq 0
Subtract 10x from both sides.
2x^{2}+10x\leq 0
Multiply the inequality by -1 to make the coefficient of the highest power in -2x^{2}-10x positive. Since -1 is negative, the inequality direction is changed.
2x\left(x+5\right)\leq 0
Factor out x.
x+5\geq 0 x\leq 0
For the product to be ≤0, one of the values x+5 and x has to be ≥0 and the other has to be ≤0. Consider the case when x+5\geq 0 and x\leq 0.
x\in \begin{bmatrix}-5,0\end{bmatrix}
The solution satisfying both inequalities is x\in \left[-5,0\right].
x\geq 0 x+5\leq 0
Consider the case when x+5\leq 0 and x\geq 0.
x\in \emptyset
This is false for any x.
x\in \begin{bmatrix}-5,0\end{bmatrix}
The final solution is the union of the obtained solutions.