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 are the values of \(x\) that make each factor equal to zero. We have the following factors:
- \(x + 1 = 0 \Rightarrow x = -1\)
- \(x - 3 = 0 \Rightarrow x = 3\)
- \(x - 5 = 0 \Rightarrow x = 5\)
- \(x - 4 = 0 \Rightarrow x = 4\)
- \(x - 2 = 0 \Rightarrow x = 2\)

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

### Step 2: Determine the sign of the expression in each interval
We will test the sign of the expression in the intervals defined by these critical points:
1. \( (-\infty, -1) \)
2. \( (-1, 2) \)
3. \( (2, 3) \)
4. \( (3, 4) \)
5. \( (4, 5) \)
6. \( (5, \infty) \)

### Step 3: Test each interval
- **Interval \( (-\infty, -1) \)**: Choose \(x = -2\)
\[ (-2 + 1)(-2 - 3)^2(-2 - 5)(-2 - 4)^2(-2 - 2) = (-1)(25)(-7)(36)(-4) > 0 \]

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

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

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

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

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

### Step 4: Determine where the inequality is satisfied
From our tests, we find that the expression is negative in the intervals:
- \( (2, 3) \)
- \( (4, 5) \)

### Step 5: Exclude critical points
We must exclude \(x = 3\) and \(x = 4\) since the factors \((x - 3)^2\) and \((x - 4)^2\) cannot be zero in the inequality.

### Step 6: Find the largest integer satisfying the inequality
The intervals where the inequality is satisfied are:
- \(2 < x < 3\)
- \(4 < x < 5\)

The largest integer in these intervals is \(2\) (from \(2 < x < 3\)).

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