To find the value of k, we need to solve both quadratic equations and determine the conditions for the roots to be equal. Let's break down the problem into steps:

Step 1: Find the quadratic equation using -5 as a root
We are given the equation 2x² + px - 15 = 0 and told that -5 is a root.
When a number is a root of an equation, it means that when we substitute that number into the equation, it becomes zero. So, we can substitute -5 into the equation and solve for p:

2(-5)² + p(-5) - 15 = 0
50 - 5p - 15 = 0
-5p + 35 = 0
-5p = -35
p = 7

Therefore, the quadratic equation with -5 as a root is 2x² + 7x - 15 = 0.

Step 2: Find the quadratic equation using equal roots
We are given the equation p(x² + x) + k = 0 and told that it has equal roots.
For a quadratic equation to have equal roots, the discriminant (b² - 4ac) must be equal to zero. So, we need to expand the equation and set the discriminant equal to zero:

p(x² + x) + k = 0
px² + px + k = 0

Comparing this equation with the standard form of a quadratic equation ax² + bx + c = 0, we can determine that a = p, b = p, and c = k.

The discriminant formula is b² - 4ac. Substituting the values, we have:

(p)² - 4(p)(k) = 0
p² - 4pk = 0

Step 3: Solve for k
We can use the quadratic equation we found in Step 1 to substitute for p in the equation from Step 2:

(7)² - 4(7)(k) = 0
49 - 28k = 0
28k = 49
k = 49/28
k = 7/4

Therefore, the value of k is 7/4.

In summary, we found that the quadratic equation with -5 as a root is 2x² + 7x - 15 = 0. We also determined that the quadratic equation p(x² + x) + k = 0 has equal roots when k = 7/4.