Final answer:
The function f(x) = -x(x+5)(x+1)(x-2) is positive for -1 < x < 0 and 0 < x < 2 within the interval -1 < x < 2, with a sign change at each zero of the function.
Step-by-step explanation:
The question asks about the sign of the function f(x) = -x(x+5)(x+1)(x-2) within the interval -1 < x < 2. To determine the sign, we should consider the zeroes of the function and the intervals they create on the number line. In this case, the zeroes are x = 0, x = -5, x = -1, and x = 2.
The value for f(x) will change sign every time we pass through a zero. We are interested in the interval between x = -1 and x = 2. Notice that we have two zeroes in this interval, x = 0 and x = -1. As x approaches from the left of -1 (but doesn't include -1 itself, as our inequality is strict), f(x) is positive because there is an even number (two) of negative factors (x and x+1). As x passes -1 and approaches 0, f(x) becomes negative, since we now have an odd number of negative factors due to the inclusion of the (x+1) term.
After x = 0, the sign will change again as we pass through the zero at x = 0. So, within the interval 0 < x < 2, f(x) is positive. Therefore, f(x) is positive for -1 < x < 0 and 0 < x < 2.