jmc

We need $-10x^2-11x+6\ge 0$. The quadratic factors as $$(2x+3)(-5x+2)\ge 0.$$ Thus the zeroes of the quadratic are at $-\frac{3}{2}$ and $\frac{2}{5}$. Since the quadratic opens downward, it is nonnegative between the zeroes. So the domain is $x\in \boxed{\left[-\frac{3}{2},\frac{2}{5}\right]}$.