Quadratic Equation:
The given quadratic equation is 2x^2 - px + q = 0.

Roots of the Quadratic Equation:
Let the roots of the quadratic equation be α and β, where α = 2+3p and β is the other root.

Sum and Product of Roots:
The sum of the roots is given by α + β = 2+3p + β = -(-p/2) = p/2.
The product of the roots is given by αβ = (2+3p)β = q/2.

Using Sum and Product of Roots:
From the given equation, we know that the sum of the roots is p/2.
So, p/2 = α + β = 2+3p + β.
Rearranging the terms, we get:
β = p/2 - 2 - 3p.

Substituting this value in the expression for αβ, we have:
αβ = (2+3p)(p/2 - 2 - 3p) = q/2.

Expanding and Simplifying:
Multiplying the terms, we get:
p^2/2 - 2p - 3p^2 - 2p - 6 - 9p - 6p^2 = q/2.

Combining like terms, we have:
(-14p^2 - 13p - 6)/2 = q/2.

Simplifying further, we get:
-14p^2 - 13p - 6 = q.

Finding the Value of p/q:
We need to find the value of p/q.
From the equation above, we can see that p/q is equal to -14p^2 - 13p - 6.

To find the value of p/q, we need to know the values of p and q. However, the values of p and q are not given in the question. Without the values of p and q, we cannot determine the exact value of p/q.

Therefore, the answer cannot be determined without additional information.

Conclusion:
The value of p/q cannot be determined without the values of p and q. Hence, the correct answer is none of the given options (1) 2, (2) 3, (3) 4, or (4) 8.