To solve this equation, we first need to find a common denominator for the fractions on the left side of the equation.

The common denominator for the fractions x+2/x-2 and x-2/x+2 is (x-2)(x+2).

Rewriting the equation with the common denominator, we get:

(x+2)(x+2)/(x-2)(x+2) + (x-2)(x-2)/(x-2)(x+2) = 2/5

Expanding the numerators, we get:

(x^2 + 4x + 4)/(x^2 - 4) + (x^2 - 4)/(x^2 - 4) = 2/5

Combining the fractions, we get:

2x^2 + 4x)/(x^2 - 4) = 2/5

Multiplying both sides by (x^2 - 4) to get rid of the denominator, we get:

2x^2 + 4x = 2/5(x^2 - 4)

Expanding the right side, we get:

2x^2 + 4x = 2/5x^2 - 8/5

Multiplying both sides by 5 to get rid of the fraction, we get:

10x^2 + 20x = 2x^2 - 8

Rearranging the equation, we get:

8x^2 - 20x + 8 = 0

Dividing by 4 to simplify the equation, we get:

2x^2 - 5x + 2 = 0

This is now a quadratic equation that can be solved using the quadratic formula or by factoring. We can factor the equation as:

(2x - 1)(x - 2) = 0

Setting each factor to zero and solving for x, we get:

2x - 1 = 0
2x = 1
x = 1/2

x - 2 = 0
x = 2

Therefore, the solutions to the equation x + 2/x - 2 + x - 2/x + 2 = 2/5 are x = 1/2 and x = 2.