To solve the inequality x^2 - 4x - 3 > 0, we first factor the quadratic equation into (x - 3)(x + 1) > 0. This inequality holds true when both factors are positive or when both factors are negative. Both factors are positive when x > 3 and x > -1, which simplifies to x > 3. Both factors are negative when x < 3 and x < -1, which simplifies to x < -1. Therefore, the solution to the inequality is x < -1 or x > 3.