How do I explain that a polynomial time reduction is in fact polynomial time?
Your reduction indeed shows that $QuadSAT\in NP$, since you showed a reduction to a problem in NP.
Perhaps an easier way to have done this, is to simply show that there is a "short witness" for the membership of a formula in $QuadSAT$.
In order to show NP-hardness, you must show a reduction from an NP-hard problem. A good candidate for that is $SAT$ (or $3SAT$).