Polynomial in two variables

Question

In the ring of polynomials of two variables $\mathbb{C}[x,y]$, the polynomial $f(x,y)=x^2+y^2+1$ is irreducible ?

Answer 1

Hint: Suppose $f(x,y) $ is reducible then $f(x,y)=g(x,y)h(x,y)$ where $\deg f(x,y)=\deg g(x,y)+\deg h(x,y)=2$ then $\deg g(x,y)=\deg h(x,y)=1$ this mean that $g(x,y)$ or $h(x,y)$ is of the form $ax+by+c$ $\Rightarrow$ make the product we have an absurd.