Could someone explain what I am missing in regard to this Wronskian?

Could someone explain what I am missing in regard to this Wronskian?

The Wronskian of $f$ and $g$ is $t^2e^t$. If $f(t)=t$ what is $g$?
Obviously, this reduces to the linear ODE $$g' + \dfrac1t g = t^2e^t$$
However, by tabular integration I arrive at a RHS of $e^tt^2 - 2te^t +
2e^t + c$.
But, the text asserts that this is equal to $te^t +ct$. How can the middle
two terms collapse if the first one is multiplied by $t$ and second is
not? That is, how is it that $t^2e^t - 2te^t + 2e^t = t^2e^t$?
Would someone please explain this to me? Thanks in advance.