What about
factoring it we have,
g(x) = (x - 2)(x^2 + 2x + 4)/(x^2 + 1),
@ x = 2; g(x) vanishes ..... this factor (x^2 + 2x + 4) has no real root,
the discriminant is b^2 - 4ac = 2^2 -4(1)(4) = -12 < 0.
The graph below confirms it, x = 2 is the only ZERO of the function