To solve the problem, we need to find the value of \( OA \cdot OB \cdot OC \cdot OD \), where \( O \) is the origin and the points \( A, B, C, D \) are the intersection points of the line \( y = \sqrt{3}x \) and the curve given by the equation:
\[
x^4 + ax^2y + bxy + 2x + dy + 6 = 0
\]
### Step 1: Substitute the line equation into the curve equation
We start by substituting \( y = \sqrt{3}x \) into the curve equation:
\[
x^4 + ax^2(\sqrt{3}x) + bxy + 2x + d(\sqrt{3}x) + 6 = 0
\]
This simplifies to:
\[
x^4 + a\sqrt{3}x^3 + b\sqrt{3}x^2 + 2x + d\sqrt{3}x + 6 = 0
\]
### Step 2: Rearranging the equation
Rearranging the equation gives us:
\[
x^4 + a\sqrt{3}x^3 + (b\sqrt{3} + 2 + d\sqrt{3})x^2 + 6 = 0
\]
### Step 3: Identify the roots
Let the roots of this polynomial be \( x_1, x_2, x_3, x_4 \). Since the line intersects the curve at four points, we can denote the coordinates of these points as follows:
- \( A(x_1, \sqrt{3}x_1) \)
- \( B(x_2, \sqrt{3}x_2) \)
- \( C(x_3, \sqrt{3}x_3) \)
- \( D(x_4, \sqrt{3}x_4) \)
### Step 4: Calculate distances from the origin
The distance from the origin \( O(0, 0) \) to point \( A \) is:
\[
OA = \sqrt{x_1^2 + (\sqrt{3}x_1)^2} = \sqrt{x_1^2 + 3x_1^2} = \sqrt{4x_1^2} = 2|x_1|
\]
Similarly, for points \( B, C, \) and \( D \):
\[
OB = 2|x_2|, \quad OC = 2|x_3|, \quad OD = 2|x_4|
\]
### Step 5: Calculate the product \( OA \cdot OB \cdot OC \cdot OD \)
Now we can compute the product:
\[
OA \cdot OB \cdot OC \cdot OD = (2|x_1|) \cdot (2|x_2|) \cdot (2|x_3|) \cdot (2|x_4|) = 16 |x_1 x_2 x_3 x_4|
\]
### Step 6: Use Vieta's formulas
According to Vieta's formulas, the product of the roots \( x_1 x_2 x_3 x_4 \) of the polynomial \( x^4 + a\sqrt{3}x^3 + (b\sqrt{3} + 2 + d\sqrt{3})x^2 + 6 = 0 \) is given by:
\[
x_1 x_2 x_3 x_4 = \frac{6}{1} = 6
\]
### Step 7: Final calculation
Substituting back into our product:
\[
OA \cdot OB \cdot OC \cdot OD = 16 \cdot 6 = 96
\]
Thus, the final answer is:
\[
\boxed{96}
\]
