(Х-1)*(х+3)^2*(х-2)<0

To solve the inequality \((x - 1)(x + 3)^2(x - 2) < 0\), we will first identify the roots of the expression and analyze the sign of the product across the intervals defined by these roots.

Step 1: Find the roots of the equation.

The roots of the expression \((x - 1)(x + 3)^2(x - 2) = 0\) are:

• \(x = 1\)
• \(x = -3\) (with multiplicity 2)
• \(x = 2\)

Step 2: Determine the intervals.

The roots divide the real number line into the following intervals:

1. \((-\infty, -3)\)
2. \((-3, 1)\)
3. \((1, 2)\)
4. \((2, \infty)\)

Step 3: Test the sign of the expression in each interval.

Let's pick a test point from each interval and evaluate the expression.

1. Interval \((-\infty, -3)\): Choose \(x = -4\) \[ (-4 - 1)(-4 + 3)^2(-4 - 2) = (-5)(-1)^2(-6) = -5 \cdot 1 \cdot -6 = 30 > 0 \] (Positive)

2. Interval \((-3, 1)\): Choose \(x = 0\) \[ (0 - 1)(0 + 3)^2(0 - 2) = (-1)(3^2)(-2) = -1 \cdot 9 \cdot -2 = 18 > 0 \] (Positive)

3. Interval \((1, 2)\): Choose \(x = 1.5\) \[ (1.5 - 1)(1.5 + 3)^2(1.5 - 2) = (0.5)(4.5^2)(-0.5) = 0.5 \cdot 20.25 \cdot -0.5 = -5.0625 < 0 \] (Negative)

4. Interval \((2, \infty)\): Choose \(x = 3\) \[ (3 - 1)(3 + 3)^2(3 - 2) = (2)(6^2)(1) = 2 \cdot 36 \cdot 1 = 72 > 0 \] (Positive)

Step 4: Analyze the roots and finding the solution.

Now we know the sign of the expression in each interval:

• \((-\infty, -3)\): Positive
• \((-3, 1)\): Positive
• \((1, 2)\): Negative
• \((2, \infty)\): Positive

The expression is negative in the interval \((1, 2)\).

Step 5: Consider endpoints.

At \(x = 1\) and \(x = 2\), the expression equals 0:

• \((x - 1) = 0\) at \(x = 1\) (neither included since we need \(< 0\))
• \((x - 2) = 0\) at \(x = 2\) (neither included since we also need \(< 0\))

At \(x = -3\), the expression is also 0 but is a root of even multiplicity, so it does not change sign at this point. Thus it's not included.

Final solution: \[ \boxed{(1, 2)} \] is the solution to the inequality \((x - 1)(x + 3)^2(x - 2) < 0\).