To solve this equation, we first need to simplify both sides by finding a common denominator:

Left side:
x + 2/2 + 2/x + 2
= (x^2 + 2(x+2) + 4) / (2(x + 2))
= (x^2 + 2x + 4) / (2x + 4)
= (x^2 + 2x + 4) / 2(x + 2)

Right side:
x + 2/3 + 3/x + 2
= (3x + 6 + 2(3)) / 3x + 6
= (3x + 12) / 3(x + 2)
= (3(x + 4)) / 3(x + 2)
= (x + 4) / (x + 2)

Now, our equation becomes:
(x^2 + 2x + 4) / 2(x + 2) = (x + 4) / (x + 2)

To solve for x, we can cross multiply:
(x^2 + 2x + 4)(x + 2) = 2(x + 4)(x + 2)
x^3 + 2x^2 + 4x + 2x^2 + 4x + 8 = 2(x^2 + 6x + 8)
x^3 + 4x^2 + 8x + 8 = 2x^2 + 12x + 16
x^3 + 2x^2 - 4x - 8 = 0

This equation can be further simplified by factoring or using another method to find the roots.