If you meant 2x2+x+6 as the quadratic, there are no real roots or zeroes, because the only roots are complex, and I don't think Algebra 1 curriculum includes complex numbers.
If you meant 2x2+x-6 then this is the same as (2x-3)(x+2), and the zeroes are x=3/2 and -2.
If this question has been wrongly categorised, and you understand complex numbers then one way of solving the quadratic (finding its zeroes or roots) is:
2x2+x+6=0,
2x2+x=-6,
x2+½x=-3,
x2+½x+1/16=-3+1/16=-47/16,
(x+¼)2=-47/16,
x+¼=±¼√(-47)=±¼i√47,
x=-¼±¼i√47. (i is the imaginary √(-1).)
(Just as "ordinary" numbers can be represented on a number line, which is one-dimensional, complex numbers are treated as if they belong to a number plane (two dimensions). The second dimension is called the imaginary dimension, while the first dimension is the real dimension of ordinary numbers.)