To start with, you must exclude the roots that are implied by the factors: (x + 3) ≠ 0 (x - 6) ≠ 0 (x + 2) ≠ 0 This is because the factors are the three places where the product actually becomes equal to 0. Therefore: x ≠ -3 x ≠ 6 x ≠ -2 This means that there are four domains for the function that excludes the three values,-3, -2, and 6: -∞ < x < -3 -3 < x < -2 -2 < x < 6 6 < x < ∞ Let's look at the first domain, -∞ < x < -3, and see what the sign of each of the factors is: (x + 3) is negative (x + 2) is negative (x - 6) is negative The product of an odd number of negative numbers is negative. Therefore, the product is less than zero for the first domain. Let's look at the second domain, -3 < x < -2, and see what the sign of each of the factors is: (x + 3) is positive (x + 2) is negative (x - 6) is negative The product of an even number of negative numbers with any number of positive numbers is a positive number. Therefore, the product is greater than zero for the second domain. Let's look at the third domain, -2 < x < 6: (x + 3) is positive (x + 2) is positive (x - 6) is negative The product of an odd number of negative numbers with any number of positive numbers is negative. Therefore, the product is less than zero for the third domain. Finally the fourth domain, 6 < x < ∞ (x + 3) is positive (x + 2) is positive (x - 6) is positive The product of any number of positive number is positive. Therefore, the product is always greater than zero for the fourth domain. The domains where the product is less than zero are: -∞ < x < -3 and -2 < x < 6

At -oo it's -oo. At +oo it's +oo And it passes through the X axis three times in -3,-2 and +6. . So x in (-oo,-3) u (-2,6) . . It looks something like the violet one on this picture: http://upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Hermite_polynomial.svg/720px-Hermite_polynomial.svg.png