Final answer:
The solution set for the inequality (x + 1)(x - 3)(x - 7)(x + 6) > 0 is option d) (-∞, -6) U (-1, 3) U (7, ∞), which is found by testing intervals around the zeros of each factor.
Step-by-step explanation:
To solve the inequality (x + 1)(x - 3)(x - 7)(x + 6) > 0, we first identify critical points where each factor is zero, which are x = -6, -1, 3, and 7. Next, we test values from intervals determined by these points to see where the product of the four factors is positive. The critical points divide the number line into five intervals: (-\(\infty\), -6), (-6, -1), (-1, 3), (3, 7), and (7, \(\infty\)). By testing a value from each interval, we find that the product is positive in the intervals (-\(\infty\), -6), (-1, 3), and (7, \(\infty\)). Therefore, the solution set for the inequality is d) (-\(\infty\), -6) U (-1, 3) U (7, \(\infty\)).