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

Our equation is:

(2x + 1)/(x - 2) + (x - 3)/(x + 2) = (x^2 + 16)/(x^2 - 4)

The common denominator here is (x - 2)(x + 2) since (x^2 - 4) = (x - 2)(x + 2)

Using the common denominator, we rewrite the equation as:

[(2x + 1)(x + 2) + (x - 3)(x - 2)]/(x - 2)(x + 2) = (x^2 + 16)/(x^2 - 4)

Expanding the numerators gives:

(2x^2 + 4x + x - 3)/(x^2 - 4) = (x^2 + 16)/(x^2 - 4)

Simplifying the fractions gives:

(2x^2 + 5x - 3)/(x^2 - 4) = (x^2 + 16)/(x^2 - 4)

Since the denominators are the same, we can equate the numerators:

2x^2 + 5x - 3 = x^2 + 16

Rearranging the terms and setting the equation to zero gives:

2x^2 + 5x - x^2 - 16 - 3 = 0

x^2 + 5x - 19 = 0

This is a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. The solutions for x are approximately:

x = -3.38 or x = 2.38

Therefore, the solutions for the equation are x = -3.38 or x = 2.38.