Analysis of the Equation:
The given equation is a quartic equation, which can be written as:
x^4 - 2x^2 + 6x - 2 = 0

To find the number of distinct real roots of this equation, we can use the concept of the discriminant.

Discriminant:
The discriminant is a mathematical term used to determine the nature of the roots of a quadratic or quartic equation. For a quartic equation of the form ax^4 + bx^3 + cx^2 + dx + e = 0, the discriminant is given by:
Δ = b^2c^2 - 4ac^3 - 4b^3d - 27a^2d^2 + 18abcd

If the discriminant is positive, the equation has two pairs of distinct real roots.
If the discriminant is zero, the equation has two pairs of equal real roots.
If the discriminant is negative, the equation has two pairs of complex roots.

Calculating the Discriminant:
For the given equation, a = 1, b = 0, c = -2, d = 6, and e = -2.
Plugging these values into the discriminant formula, we have:
Δ = (0)^2(-2)^2 - 4(1)(-2)^3 - 4(0)^3(6) - 27(1)^2(6)^2 + 18(1)(0)(-2)(6)
= 0 - 4(-8) - 0 - 27(36) + 0
= 32 - 972
= -940

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Conclusion:
The equation x^4 - 2x^2 + 6x - 2 = 0 has no distinct real roots.