Find the number of integral values of 'a' for which `ax^2 - (3a + 2)x + 2(a + 1) < 0, a != 0` holds exactly four integral value of `x.`

Answer

To find the number of integral values of 'a' for which the inequality \( ax^2 - (3a + 2)x + 2(a + 1) < 0 \) holds for exactly four integral values of \( x \), we can follow these steps:

### Step 1: Analyze the Quadratic Inequality
The given quadratic inequality can be expressed in the standard form:
\[ f(x) = ax^2 - (3a + 2)x + 2(a + 1) \]
We need to determine when this quadratic function is less than zero for exactly four integral values of \( x \).

### Step 2: Determine the Nature of the Roots
For the quadratic to be less than zero for exactly four integral values of \( x \), it must have two distinct real roots, say \( \alpha \) and \( \beta \), such that the interval between them contains exactly four integers. This means:
\[ \lfloor \beta \rfloor - \lceil \alpha \rceil = 3 \]
This implies that the roots must be spaced such that the distance between them is slightly more than 4.

### Step 3: Calculate the Discriminant
For the quadratic to have two distinct real roots, the discriminant must be positive:
\[ D = b^2 - 4ac > 0 \]
Substituting \( a = a \), \( b = -(3a + 2) \), and \( c = 2(a + 1) \):
\[ D = (3a + 2)^2 - 4a \cdot 2(a + 1) > 0 \]
Calculating this gives:
\[ D = (3a + 2)^2 - 8a(a + 1) \]
\[ = 9a^2 + 12a + 4 - 8a^2 - 8a \]
\[ = a^2 + 4a + 4 > 0 \]

### Step 4: Factor the Discriminant
The expression \( a^2 + 4a + 4 \) can be factored as:
\[ (a + 2)^2 > 0 \]
This inequality holds for all \( a \) except \( a = -2 \).

### Step 5: Determine the Roots
Next, we need to find the roots of the quadratic:
\[ ax^2 - (3a + 2)x + 2(a + 1) = 0 \]
Using the quadratic formula:
\[ x = \frac{-(3a + 2) \pm \sqrt{D}}{2a} \]
The roots will be:
\[ x_1, x_2 = \frac{3a + 2 \pm \sqrt{D}}{2a} \]

### Step 6: Finding the Integral Values
For the roots \( x_1 \) and \( x_2 \) to have exactly four integers between them, we need:
\[ \lfloor x_2 \rfloor - \lceil x_1 \rceil = 3 \]
This condition will depend on the specific values of \( a \).

### Step 7: Count the Integral Values of \( a \)
To satisfy the condition that \( a \neq 0 \) and \( a \neq -2 \), we can check integer values of \( a \) around the critical points. The quadratic nature of the expression suggests that there will be an infinite number of integral values for \( a \) except for the excluded values.

### Conclusion
The number of integral values of \( a \) for which the inequality holds for exactly four integral values of \( x \) is infinite, excluding \( a = 0 \) and \( a = -2 \).