The real roots of the equation ${x^2} + 5|x| + \,\,4 = 0$ are

How many positive real numbers $x$ satisfy the equation $x^3-3|x|+2=0$ ?

Let $\mathrm{S}$ be the set of positive integral values of $a$ for which $\frac{\mathrm{ax}^2+2(\mathrm{a}+1) \mathrm{x}+9 \mathrm{a}+4}{\mathrm{x}^2-8 \mathrm{x}+32}<0, \forall \mathrm{x} \in \mathbb{R}$. Then, the number of elements in $\mathrm{S}$ is :

Sum of the solutions of the equation $\left[ {{x^2}} \right] - 2x + 1 = 0$ is (where $[.]$ denotes greatest integer function)

The set of values of $x$ which satisfy $5x + 2 < 3x + 8$ and $\frac{{x + 2}}{{x - 1}} < 4,$ is

Exact set of values of $a$ for which ${x^3}(x + 1) = 2(x + a)(x + 2a)$ is having four real solutions is