Nah, don't do it for that question. Like you said, messy polynomial.
Basically, they've already given you the velocity with i and j components. And you have the direction that its going in, in i and j components. So it's like equating coefficients. Say the velocity was k(i-j) = ki - kj. Equating coeffs:
k = 2q^2 - 3
-k = q+2
k = -(q+2)
Now, since they're both equal to 'k' we can equate them, thus:
2q^2 - 3 = -(q+2)
etc...
Say it was k(2i+3j), then you can do the same thing:
2k = 2q^2 - 3
3k = q+2
You can divide the first by 2 and the second by 3, and then equate.