The number of integer values of a for which `x^2+ 3ax + 2009 = 0` has two integer roots is :

Answer

To solve the problem of finding the number of integer values of \( a \) for which the equation \( x^2 + 3ax + 2009 = 0 \) has two integer roots, we can follow these steps:

### Step 1: Identify the roots
For a quadratic equation \( ax^2 + bx + c = 0 \), the sum and product of the roots can be expressed as:
- Sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- Product of the roots \( \alpha \beta = \frac{c}{a} \)

In our case, the equation is:
\[ x^2 + 3ax + 2009 = 0 \]
Here, \( a = 1 \), \( b = 3a \), and \( c = 2009 \).

Thus, the sum of the roots is:
\[ \alpha + \beta = -\frac{3a}{1} = -3a \]
And the product of the roots is:
\[ \alpha \beta = \frac{2009}{1} = 2009 \]

### Step 2: Set up the equations
From the above, we have:
1. \( \alpha + \beta = -3a \)
2. \( \alpha \beta = 2009 \)

### Step 3: Use the quadratic formula
The roots of the quadratic equation can also be found using the discriminant:
\[ D = b^2 - 4ac = (3a)^2 - 4 \cdot 1 \cdot 2009 = 9a^2 - 8036 \]

For the roots to be integers, the discriminant must be a perfect square:
\[ D \geq 0 \implies 9a^2 - 8036 \geq 0 \]
\[ 9a^2 \geq 8036 \]
\[ a^2 \geq \frac{8036}{9} \]

### Step 4: Calculate the minimum value of \( a \)
Now we calculate \( \frac{8036}{9} \):
\[ 8036 \div 9 \approx 892.888\ldots \]
Thus,
\[ a^2 \geq 893 \implies a \geq \sqrt{893} \quad \text{or} \quad a \leq -\sqrt{893} \]
Calculating \( \sqrt{893} \) gives approximately \( 29.9 \). Therefore, the integer values of \( a \) must satisfy:
\[ a \geq 30 \quad \text{or} \quad a \leq -30 \]

### Step 5: Find integer values of \( a \)
The integer values of \( a \) can be:
- \( a = 30, 31, 32, \ldots \) (and so on)
- \( a = -30, -31, -32, \ldots \) (and so on)

### Step 6: Determine bounds
To find the maximum integer values, we need to check the limits:
- For \( a \geq 30 \), there is no upper limit.
- For \( a \leq -30 \), there is also no lower limit.

### Conclusion
The integer values of \( a \) are \( a \geq 30 \) and \( a \leq -30 \). Therefore, the total number of integer values of \( a \) is infinite.