Find the largest integral `x` which satisfies the following inequality:
`(x+1)(x-3)^(2) (x-5)(x-4)^(2)(x-2) lt 0`

Answer

To solve the inequality \((x+1)(x-3)^{2}(x-5)(x-4)^{2}(x-2) < 0\), we will follow these steps:

### Step 1: Identify the critical points
The critical points occur when each factor is equal to zero. Thus, we set each factor to zero:

1. \(x + 1 = 0 \Rightarrow x = -1\)
2. \(x - 3 = 0 \Rightarrow x = 3\)
3. \(x - 5 = 0 \Rightarrow x = 5\)
4. \(x - 4 = 0 \Rightarrow x = 4\)
5. \(x - 2 = 0 \Rightarrow x = 2\)

The critical points are \(x = -1, 2, 3, 4, 5\).

### Step 2: Analyze the intervals
We will analyze the sign of the expression in the intervals defined by the critical points. The intervals are:
- \((-∞, -1)\)
- \((-1, 2)\)
- \((2, 3)\)
- \((3, 4)\)
- \((4, 5)\)
- \((5, ∞)\)

### Step 3: Test the intervals
We will choose test points from each interval to determine the sign of the expression:

1. **Interval \((-∞, -1)\)**: Choose \(x = -2\)
\[ (-2 + 1)(-2 - 3)^{2}(-2 - 5)(-2 - 4)^{2}(-2 - 2) = (-1)(25)(-7)(36)(-4) > 0 \]

2. **Interval \((-1, 2)\)**: Choose \(x = 0\)
\[ (0 + 1)(0 - 3)^{2}(0 - 5)(0 - 4)^{2}(0 - 2) = (1)(9)(-5)(16)(-2) > 0 \]

3. **Interval \((2, 3)\)**: Choose \(x = 2.5\)
\[ (2.5 + 1)(2.5 - 3)^{2}(2.5 - 5)(2.5 - 4)^{2}(2.5 - 2) = (3.5)(0.25)(-2.5)(2.25)(0.5) < 0 \]

4. **Interval \((3, 4)\)**: Choose \(x = 3.5\)
\[ (3.5 + 1)(3.5 - 3)^{2}(3.5 - 5)(3.5 - 4)^{2}(3.5 - 2) = (4.5)(0.25)(-1.5)(0.25)(1.5) < 0 \]

5. **Interval \((4, 5)\)**: Choose \(x = 4.5\)
\[ (4.5 + 1)(4.5 - 3)^{2}(4.5 - 5)(4.5 - 4)^{2}(4.5 - 2) = (5.5)(2.25)(-0.5)(0.25)(2.5) < 0 \]

6. **Interval \((5, ∞)\)**: Choose \(x = 6\)
\[ (6 + 1)(6 - 3)^{2}(6 - 5)(6 - 4)^{2}(6 - 2) = (7)(9)(1)(4)(4) > 0 \]

### Step 4: Determine the intervals where the expression is negative
From our analysis, the expression is negative in the intervals:
- \((2, 3)\)
- \((3, 4)\)
- \((4, 5)\)

### Step 5: Identify the largest integral value of \(x\)
The integral values in the intervals where the expression is negative are:
- From \((2, 3)\): No integers
- From \((3, 4)\): No integers
- From \((4, 5)\): Only \(4\) (but \(x\) cannot be equal to \(4\))

The largest integral value satisfying the inequality is \(2\) (since \(3\) and \(4\) are excluded).

### Final Answer
The largest integral \(x\) which satisfies the inequality is \(\boxed{2}\).