An irreducible polynomial

Question

Let $k$ be an algebraically closed field and $f\in k[x_0,x_1,y_0,y_1,y_2]$ be given by $$f(x_0,x_1,y_0,y_1,y_2)=y_0x_0^2+y_1x_0x_1+y_2x_1^2.$$ Why is $f$ irreducible?

Answer 1

Note that $f(x_0,x_1,y_0,y_1,y_2)$ is homogeneous of degree $2$ w.r.t. the set of indeterminates $\{x_0,x_1\}$, and homogeneous linear w.r.t. $\{y_0,y_1,y_2\}$.

So it suffices to show the de-homogenised polynomial is irreducible: $$F(X, Y_1,Y_2)=1+Y_1X+Y_2X^2 \qquad(X=\frac{x_1}{x_0},\;Y_1=\frac{y_1}{y_0},\;Y_2=\frac{y_2}{y_0}) $$ Can you take it from here?