Explanation:

To find the group velocity of the wave, we need to find the dispersion relation, which relates the wave's frequency (ω) to its wavenumber (k).

Given, ω = 3k^3 + 2k^2 + k

Differentiating with respect to k, we get:

dω/dk = 9k^2 + 4k + 1

Now, the group velocity (vg) is given by:

vg = dω/dk

Substituting the expression for dω/dk, we get:

vg = 9k^2 + 4k + 1

Therefore, the group velocity of the wave is given by the expression 9k^2 + 4k + 1.