You are looking only for real solutions, so you should restrict Solve:

Solve[y == (8 + 10 x^2 + x^4 + 6 x Sqrt[6 + x^2] + x^3 Sqrt[6 + x^2])/(8 + x^2), x, Reals] 

gives two real (not four) solution branches as expected.

You are looking only for real solutions, so you should restrict Solve:

Solve[y == (8 + 10 x^2 + x^4 + 6 x Sqrt[6 + x^2] + x^3 Sqrt[6 + x^2])/(8 + x^2), x, Reals] 

gives two real (not four) solution branches as expected.

addenum

It seems that Solve (without restricting reals) expands the soultion range by transforming(squaring the roots) the equation to (y (8 + x^2) - (8 + 10 x^2 + x^4))^2 == (6 x + x^3)^2 (6 + x^2)

ContourPlot[(y (8 + x^2) - (8 + 10 x^2 + x^4))^2 == (6 x + x^3)^2 (6 +x^2), {x, -10, 10}, {y, 0, 10}, FrameLabel -> {x, y}]