I like to let a graphing calculator help with polynomials of higher degree.
A graph of p(x) shows it has one real zero, at x=1. Dividing that out (and dividing out the scale factor of 2) reveals the remaining factor to be
.. (x -1/4)^2 +15/16
When we set this to zero, we have
.. (x -1/4)^2 +15/16 = 0
.. (x -1/4)^2 = -15/16
.. x -1/4 = i√(15/16)
and the remaining roots are
.. x = 1/4 ±i√(15/16)
.. x = (1 ±i√15)/4
The zeros are x = 1, x = (1 ±i√15)/4.
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You could use synthetic or long division to find
.. p(x)/(x -1) = 2x^2 -x +2
and then use any of several methods to determine the zeros of this are
.. x = (1 ±i√15)/4