The resultant of two polynomials $f\left(x\right)=a_{n}x^{n}+\ldots+a_{0},g\left(x\right)=b_{n}x^{m}+\ldots+b_{0}\in K\left[x\right]$ over a field K defined as $\det R\left(f,g\right)$ when $R(f,g)$ is the sylvester matrix of $f,g$:
$$R\left(f,g\right)=\begin{pmatrix}a_{n} & a_{n-1} & \cdots & a_{0} & 0 & \cdots & \cdots & 0\\ 0 & a_{n} & a_{n-1} & \cdots & a_{0} & 0 & \cdots & 0\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots\\ b_{m} & b_{m-1} & \cdots & b_{0} & 0 & \cdots & \cdots & 0\\ 0 & b_{m} & b_{m-1} & \cdots & b_{0} & 0 & \cdots & 0\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \end{pmatrix}$$
Why the rows of $R(f,g)$ represent the coefficients of $f,xf,\ldots,x^{m-1}f,g,xg,\ldots,x^{n-1}g$?
For example, how can we recognize the second row as coefficients of $xf$?
You are reading the rows backwards, it's $x^{m-1}f, x^{m-2}f, \dots, f, x^{n-1}g, \dots, g$.