Given information:
- The quadratic equation 2x^2 + px - 15 = 0 has -5 as one of its roots.
- The quadratic equation p(x^2 + X)k = 0 has equal roots.

To Find:
The value of k.

Solution:

[Step 1] Finding the quadratic equation using the given root:
Since -5 is a root of the quadratic equation 2x^2 + px - 15 = 0, we can use it to find the other root.

[Step 2] Using the sum and product of roots formulas:
We know that the sum of the roots of a quadratic equation is given by the formula:
Sum of roots = -b/a

In this case, the sum of the roots is -p/2. Since we know one root is -5, we can substitute these values into the formula:
-5 + second root = -p/2

Simplifying the equation:
second root = -p/2 + 5

We also know that the product of the roots is given by the formula:
Product of roots = c/a

In this case, the product of the roots is -15/2. We can substitute these values into the formula:
-5 * second root = -15/2

Simplifying the equation:
second root = -15/2 * -1/5

[Step 3] Finding the second root:
From the above equation, we can find the value of the second root:
second root = 3

[Step 4] Writing the quadratic equation:
Now we have both roots of the quadratic equation, -5 and 3. We can write the equation in factored form using these roots:
(x + 5)(x - 3) = 0

Expanding the equation:
x^2 + 2x - 15 = 0

Comparing this equation with the given equation, we can see that p = 2.

[Step 5] Finding the value of k:
Now we can substitute the value of p into the equation p(x^2 + X)k = 0:
2(x^2 + x)k = 0

Since the quadratic equation has equal roots, it means that the discriminant is equal to zero. The discriminant formula is given by:
Discriminant = b^2 - 4ac

In this case, a = 2, b = 1, and c = 0. Substituting these values into the formula:
1^2 - 4(2)(0) = 0

Simplifying the equation:
1 - 0 = 0

Since the discriminant is zero, the quadratic equation has equal roots. Therefore, the value of k is 0.

Answer:
The value of k is 0.