Of course you can't seem to come up with two factors
that work.  There ARE NONE!  :)

That's why they told you to COMPLETE THE SQUARE instead.

Completing the square is what you have to do when the
quadratic cannot be factored.

p²+16p-22 = 0

Isolate the variable terms by adding 22 to both sides:

   p²+16p = 22

Multiply the coefficient of p, which is 16, by  getting
16· = 8.  Then square 8, getting 64.  Add 64 to both sides 
of the equation.

p²+16p+64 = 22+64

Now the left side will factor.  Combine the numbers on the right:

(p+8)(p+8) = 86

Note that the left side has factored a special way.  Both factors are
the same, so we can write:

    (p+8)² = 86

Now we take square roots of both sides:

   √(p+8)² = ±√86

       p+8 = ±√86
   
         p = -8±√86

So there are two solutions:

Using the +:   p = -8+√86
 which is approximately 1.273618495

Using the -:   p = -8-√86 
which is approximately -17.2736185

Edwin