**Given Information:**
- The quadratic equation is given as 2x^2 + px + 15 = 0.
- The root of the equation is -3.
- We need to find the value of q for which the equation p(x^2 + x) + q = 0 has equal roots.

**Solution:**
To find the value of q, we need to determine the conditions for the roots of the quadratic equation p(x^2 + x) + q = 0 to be equal.

**Condition for Equal Roots:**
For a quadratic equation ax^2 + bx + c = 0 to have equal roots, the discriminant (b^2 - 4ac) must be equal to zero.

**Finding the Discriminant:**
Let's find the discriminant for the given equation p(x^2 + x) + q = 0.

The given equation can be rewritten as px^2 + px + qx + q = 0.

Comparing it with the standard form ax^2 + bx + c = 0, we can determine the values of a, b, and c as follows:
- a = p
- b = p + q
- c = q

The discriminant is calculated as follows:
D = (b^2 - 4ac)
= [(p + q)^2 - 4pq]

**Condition for Equal Roots:**
To have equal roots, the discriminant D must be equal to zero.

Therefore, we have the equation: (p + q)^2 - 4pq = 0.

**Substituting the Value of p:**
We know that -3 is one of the roots of the quadratic equation 2x^2 + px + 15 = 0.

Substituting x = -3 into the equation, we get:
2(-3)^2 + p(-3) + 15 = 0
18 - 3p + 15 = 0
33 - 3p = 0
3p = 33
p = 11

**Substituting the Value of p in the Equation:**
Now, substitute the value of p = 11 into the equation (p + q)^2 - 4pq = 0.

(11 + q)^2 - 4(11)(q) = 0
121 + 22q + q^2 - 44q = 0
q^2 - 22q + 121 - 44q = 0
q^2 - 66q + 121 = 0

**Finding the Value of q:**
To find the value of q, we need to solve the quadratic equation q^2 - 66q + 121 = 0.

By factoring or using the quadratic formula, we can find that q = 1/4 and q = 121 are the solutions of the equation.

However, we need to select the value of q for which the equation has equal roots.

Since the question asks for the value of q where the equation has equal roots, the correct answer is q = 1/4 (option A).