That's an important first step. You've proven that it works for n = 1. In order to use induction, we need to prove that if it's true for n-1, then it must be true for n. Let Sn be the sum through n. Then,
Sn = Sn-1 + 2n2 - 4n + 3 (based on the definition of our sum)
Sn = (n-1)/3 * (2(n-1)2 - 3(n-1) + 4) + 2n2 - 4n + 3 (based on substituting in our formula for Sn-1)
Now, all you need to do is multiply out and simplify this expression and demonstrate that it comes out to
n/3(2n2 - 3n + 4)
If you need additional help with that, feel free to let me know.