-8x^2-4x-1<0 inequation

Given the inequality:
$$- 8 x^{2} - 4 x - 1 < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$- 8 x^{2} - 4 x - 1 = 0$$
Solve:
This equation is of the form

a*x^2 + b*x + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -8$$
$$b = -4$$
$$c = -1$$
, then

D = b^2 - 4 * a * c = 

(-4)^2 - 4 * (-8) * (-1) = -16

Because D<0, then the equation
has no real roots,
but complex roots is exists.

x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or
$$x_{1} = - \frac{1}{4} - \frac{i}{4}$$
Simplify
$$x_{2} = - \frac{1}{4} + \frac{i}{4}$$
Simplify
$$x_{1} = - \frac{1}{4} - \frac{i}{4}$$
$$x_{2} = - \frac{1}{4} + \frac{i}{4}$$
Exclude the complex solutions:
This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example

x0 = 0

$$\left(-1\right) 1 - 8 \cdot 0^{2} - 4 \cdot 0 < 0$$

-1 < 0

so the inequality is always executed