SOLUTION: Find all real values of p such that 2(x+6)(x-p)\] has a minimum value of -4 over all real values of x.

Expression  2(x+6)(x-p)  represents a parabola opened upward
with the vertex  at  x= ,  half-way between x-intercepts -6 and p.


The y-value at the vertex is  

     =  = .


We want

     = -4.


It gives  

     = 8

    6 + p = +/- 

    p = -6 +/- .


ANSWER.  There are two real numbers "p" satisfying the imposed requirements.

         They are  -6 -   and  -6 + .