Edit nevermind, there doesn't seem to be an exact value from polynomials O_O
The best we can do (if we don't use the general formula for a quartic, which is "butt ugly") is to try to find quadratic factors of the form:
(x - (a + Ib))(x - (a - Ib))
I messed with this for a while, but couldn't come up with anything intelligible.
I do not think there is nice way to solve this without the quartic formula. If instead of the 7 you have a -7 it would work.
Let me explain. Some of these polynomials can be solved by manipulating them moving factors until you bring it to the form you can solve it in.
I was able to show this polynomial is irreducible over Q. Meaning we cannot achieve this with doing nice moves, i.e. moving factors around if you know what I mean.