The given quadratic equation is 2x^2 + px - 15 = 0, and we are given that -5 is one of its roots. We need to find the value of k for the quadratic equation p(x^2 + x) + k = 0 to have equal roots.

To find the value of p, we can use the fact that -5 is a root of the equation 2x^2 + px - 15 = 0. This means that if we substitute x = -5 into the equation, we should get zero.

Substituting x = -5 into the equation, we have:
2(-5)^2 + p(-5) - 15 = 0
2(25) - 5p - 15 = 0
50 - 5p - 15 = 0
-5p + 35 = 0
-5p = -35
p = 7

Therefore, the value of p is 7.

Now, let's consider the quadratic equation p(x^2 + x) + k = 0. We are given that it has equal roots, which means its discriminant is zero.

The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by the formula:
D = b^2 - 4ac

In our case, the quadratic equation is p(x^2 + x) + k = 0, so we have:
D = (1)^2 - 4p(k)

Since the discriminant is zero, we can set it equal to zero and solve for k:
(1)^2 - 4p(k) = 0
1 - 4(7)(k) = 0
1 - 28k = 0
-28k = -1
k = 1/28

Therefore, the value of k is 1/28.

In summary:
- The value of p is 7, which is found by substituting x = -5 into the equation 2x^2 + px - 15 = 0.
- The value of k is 1/28, which is obtained by setting the discriminant of the equation p(x^2 + x) + k = 0 equal to zero and solving for k.