Interval of 'a' for the given inequality

To find the interval of 'a' for which the inequality (x^2 - ax - 1 - 2a^2) > 0 holds true for all real numbers x, we need to analyze the properties of the quadratic expression.

Step 1: Analyzing the discriminant
The discriminant of the quadratic expression (x^2 - ax - 1 - 2a^2) is given by D = b^2 - 4ac, where a = 1, b = -a, and c = -1 - 2a^2.

Substituting these values, we get:
D = (-a)^2 - 4(1)(-1 - 2a^2)
= a^2 + 4 + 8a^2
= 9a^2 + 4

Step 2: Analyzing the roots of the quadratic expression
The quadratic expression (x^2 - ax - 1 - 2a^2) can be factorized as (x - α)(x - β), where α and β are the roots.

Using Vieta's formulas, we have:
α + β = a
αβ = -1 - 2a^2

Step 3: Analyzing the sign of the quadratic expression
For the quadratic expression to be greater than zero, its graph should be above the x-axis. This can be determined by analyzing the sign of the quadratic expression for different intervals of x.

Case 1: D > 0
If the discriminant D is greater than zero, the quadratic expression has two distinct real roots. In this case, we need to analyze the sign of the expression in the intervals (-∞, α), (α, β), and (β, ∞).

Case 2: D = 0
If the discriminant D is equal to zero, the quadratic expression has two identical real roots. In this case, we need to analyze the sign of the expression in the intervals (-∞, α) and (α, ∞).

Case 3: D < />
If the discriminant D is less than zero, the quadratic expression does not have real roots. In this case, the expression will have a constant sign for all real numbers x.

Step 4: Determining the interval of 'a'
To satisfy the given inequality (x^2 - ax - 1 - 2a^2) > 0 for all real numbers x, we need the quadratic expression to be positive for all x-values. Hence, we need to ensure that the quadratic expression satisfies the following conditions:

1. If D > 0, the quadratic expression should be positive for all x-values.
2. If D = 0, the quadratic expression should be positive for all x-values.
3. If D < 0,="" the="" quadratic="" expression="" should="" be="" negative="" for="" all="" />

Final Interval:
By analyzing the above conditions, we can determine the interval of 'a' as follows:

1. If D > 0, then the quadratic expression has two distinct real roots. In this case, the interval of 'a' will be determined by the condition that the