In my lecture notes we have the following:
We have that $f(x, y), g(x, y) \in \mathbb{C}[x, y]$
$$f(x,y)=a_0(y)+a_1(y)x+ \dots +a_n(y)x^n \\ g(x, y)=b_0(y)+b_1(y)x+ \dots +b_m(y)x^m$$
The resultant is defined in the following way:
$$Res(f,g)(y)=det\begin{bmatrix} a_0(y) & a_1(y) & \dots & a_n(y) & 0 & 0 & \dots & 0 \\ 0 & a_0(y) & \dots & a_{n-1}(y) & a_n(y) & 0 & \dots & 0 \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ 0 & 0 & a_0(y) & \dots & \dots & \dots & \dots & a_n(y) \\ b_0(y) & b_1(y) & b_2(y) & \dots & b_m(y) & 0 & \dots & 0 \\ 0 & b_0(y) & b_1(y) & \dots & b_m(y) & \dots & \dots & 0 \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ 0 & 0 & 0 & b_0(y) & \dots & \dots & \dots & b_m(y) \end{bmatrix}$$
I haven't understood how the resultant is defined.
For example when we have $f(x, y)=y^2+x^2$ and $g(x, y)=y+x$, we have that $a_0(y)=y^2, a_1(y)=0, a_2(y)=1, b_0(y)=y, b_1(y)=1$.
How do we create the matrix?
In your case the matrix is given by:
$$\begin{pmatrix}y^2 & 0 & 1 \\ y & 1 & 0 \\ 0 & y & 1 \end{pmatrix} $$ In general, note that you have $m$ lines of $a$'s and $n$ lines of $b$'s and most importantly that the final result need to be and $(n+m) \times (n+m)$ matrix. Put differently, to the first line of $a$'s pad $m-1$ entries of $0$, and to the first line of $b$' pad $n-1$.
This matrix is called "Sylvester matrix" which should lead you to further examples.