Find all values of $p$ such that$$2(x+4)(x-2p)$$has a minimum value of $-18$ over all real values of $x$. (in other words, we cannot have $x$ be nonreal.)

Using the distributive properly of the expressions, we can simplify the given complicated expression into,
    (2)(x + 4)(x - 2p)
    (2x + 8)(x - 2p)
      2x² + 4px + 8x + 16p
Grouping of like terms,
    2x² + (4p + 8)x + 16p

Then, we derive the equation and equate to zero to get the minimum,
    4x + (4p + 8) = 0

It is given that the minimum value for x is equal to -18. Substitute -18 to the x from the above equation,
   4(-18) + (4p + 8) = 0
Simplifying,
  4p  - 64 = 0,
   p = 16

The value of p is then equal to 16

ANSWER: p = 16