Multiply the inequality by -1 to make the coefficient of the highest power in $-5x^{2}-3x+2$ positive. Since $-1$ is negative, the inequality direction is changed.
$$5x^{2}+3x-2\geq 0$$
To solve the inequality, factor the left hand side. Quadratic polynomial can be factored using the transformation $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$, where $x_{1}$ and $x_{2}$ are the solutions of the quadratic equation $ax^{2}+bx+c=0$.
$$5x^{2}+3x-2=0$$
All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. Substitute $5$ for $a$, $3$ for $b$, and $-2$ for $c$ in the quadratic formula.
For the product to be $≥0$, $x-\frac{2}{5}$ and $x+1$ have to be both $≤0$ or both $≥0$. Consider the case when $x-\frac{2}{5}$ and $x+1$ are both $≤0$.
$$x-\frac{2}{5}\leq 0$$ $$x+1\leq 0$$
The solution satisfying both inequalities is $x\leq -1$.
$$x\leq -1$$
Consider the case when $x-\frac{2}{5}$ and $x+1$ are both $≥0$.
$$x+1\geq 0$$ $$x-\frac{2}{5}\geq 0$$
The solution satisfying both inequalities is $x\geq \frac{2}{5}$.
$$x\geq \frac{2}{5}$$
The final solution is the union of the obtained solutions.
