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

The common denominator for x^2-4 and x+2 is (x-2)(x+2) = x^2 - 4.

Multiplying both sides of the equation by x^2-4 to eliminate the denominators, we get:

(x^2 + 2x - 8) = 7(x-2)

Expanding both sides of the equation, we have:

x^2 + 2x - 8 = 7x - 14

Rearranging the terms on the right side of the equation, we get:

x^2 + 2x - 8 = 7x - 14

Combining like terms, we get:

x^2 - 5x + 6 = 0

Now, we need to factor this quadratic equation:

(x - 2)(x - 3) = 0

Setting each factor equal to zero:

x - 2 = 0
x = 2

x - 3 = 0
x = 3

Therefore, the solutions to the equation are x = 2 and x = 3.