How to shorten dot product

Question

I would like to shorten a dot/scalar product: $$f(s)=sP_1+s^2P_2+\big((P_2-P_1)^TsN_1\big)N_1$$ Here $s$ are scalars, $P$ are points and $N$ are unit normal vectors in $R^3$. The function $f(s)$ returns a point. Of course $$(P_2-P_1)^TsN_1$$ is a dot product. How can I rewrite the equation to show it is a quadratic function e.g. in polynomial form? That is why I would like to rewrite the dot product.