What does the fundamental theorem of algebra illustrate?

Step-by-step explanation:

The "fundamental theorem of algebra" tells you a polynomial has a number of roots equal to its degree. Those roots are not always rational or real, but are always members of the set of complex numbers.

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The given equation can be put into vertex form as ...

  2(x -1)^2 -3 = 0

It will have roots ...

  1 ± √(3/2) = (2±√6)/2

If you don't like to mess with translating a quadratic to vertex form, you can use the quadratic formula:

  x = (-b±√(b²-4ac))/(2a) . . . . . for a=2, b=-4, c=-1

  x = (4 ± √(16+8))/(2·2) = (4 ± 2√6)/4 = (2±√6)/2