x^2(x-2)(x+3)^2

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x^2(x-2)(x+3)^2<0

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Identify Critical Points: Identify the critical points of the inequality by setting each factor equal to zero. x^2 = 0 gives x = 0 (x - 2) = 0 gives x = 2 (x + 3)^2 = 0 gives x = -3Plot on Number Line: Plot the critical points on a number line and determine the intervals to test. The critical points divide the number line into four intervals: (-∞, -3), (-3, 0), (0, 2), and (2, ∞).Choose Test Points: Choose test points from each interval and plug them into the inequality to determine the sign of the expression in that interval. For (-∞, -3), choose x = -4: (-4)^2(-4 - 2)(-4 + 3)^2 > 0, which is positive. For (-3, 0), choose x = -1: (-1)^2(-1 - 2)(-1 + 3)^2 < 0, which is negative. For (0, 2), choose x = 1: (1)^2(1 - 2)(1 + 3)^2 < 0, which is negative. For (2, ∞), choose x = 3: (3)^2(3 - 2)(3 + 3)^2 > 0, which is positive.Determine Satisfied Intervals: Determine the intervals where the inequality is satisfied. The inequality is satisfied in the intervals where the expression is negative, which are (-3, 0) and (0, 2).Consider Critical Points: Consider the behavior at the critical points. Since the inequality is strict (<0), the critical points themselves are not included in the solution.Combine Intervals: Combine the intervals to express the solution. The solution to the inequality is the union of the intervals where the expression is negative, which is (-3, 0) U (0, 2).

x^2(x-2)(x+3)^2<0

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