Let $f(x, y) = x^2 y + y^2 e^x$. Consider the second-order Taylor polynomial $P_2(x, y)$ centered at $(0, 1)$. Which of the following coefficients correspond to the terms $(x-0)(y-1)$ and $\frac{1}{2}(y-1)^2$ respectively in the expansion $f(0, 1) + f_x(0, 1)x + f_y(0, 1)(y-1) + \frac{1}{2}f_{xx}(0, 1)x^2 + f_{xy}(0, 1)x(y-1) + \frac{1}{2}f_{yy}(0, 1)(y-1)^2$?