Final answer:
The correct answer is option a. By analyzing the inequality (x + 6)(x + 1) > 0 and testing intervals, it is determined that the solution on the number line includes the intervals x < -6 and x > -1, with open circles at -6 and -1.
Step-by-step explanation:
The correct answer is option a. To solve the inequality (x + 6)(x + 1) > 0, we need to find the values of x where the product of these two factors is positive. This product will be positive if both factors are positive or both factors are negative. The factors switch from negative to positive or vice versa at their respective zeros, which are x = -6 and x = -1. This gives us three intervals to test: x < -6, -6 < x < -1, and x > -1.
By selecting a test point from each interval and substituting it into the inequality, we find that the inequality holds true when x < -6 and x > -1. Therefore, we graph the solution on the number line as open circles at x = -6 and x = -1 (because the inequality is strict and does not include these points) and shade the regions to the left of x = -6 and to the right of x = -1.