To solve the equation, let's expand the expressions on both sides:

(2x-1)^2 - (x+3)^2 = 0

Expanding the squares:

(4x^2 - 4x + 1) - (x^2 + 6x + 9) = 0

Now, let's simplify the equation by combining like terms:

4x^2 - 4x + 1 - x^2 - 6x - 9 = 0

Combining like terms:

3x^2 - 10x - 8 = 0

Now, we have a quadratic equation. To solve it, we can either factor it or use the quadratic formula. Let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 3, b = -10, and c = -8. Substituting these values into the formula:

x = (-(-10) ± √((-10)^2 - 4(3)(-8))) / (2(3))

Simplifying:

x = (10 ± √(100 + 96)) / 6

x = (10 ± √196) / 6

x = (10 ± 14) / 6

Now, we have two possible solutions:

x = (10 + 14) / 6 = 24 / 6 = 4

x = (10 - 14) / 6 = -4 / 6 = -2/3

Therefore, the solutions to the equation (2x-1)^2 - (x+3)^2 = 0 are x = 4 and x = -2/3.