Let's simplify the equation:

(1+x^2)^2 + 0.5(1+x^2) - 5 = 0
Expand the squared term:
1 + 2x^2 + x^4 + 0.5(1+x^2) - 5 = 0
Combine like terms:
x^4 + 2.5x^2 - 3.5 = 0

This is a quadratic equation in terms of x^2. Let's solve for x^2 using the quadratic formula:

x^2 = (-2.5 ± √(2.5^2 - 41(-3.5))) / 2*1
x^2 = (-2.5 ± √(6.25 + 14)) / 2
x^2 = (-2.5 ± √20.25) / 2

x^2 = (-2.5 ± 4.5) / 2
x^2 = (2 or -7) / 2

Therefore, x^2 = 1 or x^2 = -3.5

Taking the square root of both sides:

x = ±1 or x = ±√(-3.5)

So the solutions to the equation are x = 1, x = -1, x = i√3.5, or x = -i√3.5.