abstract algebra - Do polynomials in two variables always factor in linear terms?

Consider a polynomial of one variable over $\Bbb C$:
$$p(x)=a_0+a_1x+\cdots+a_nx^n,\quad a_i\in\Bbb C.$$
We know from the Fundamental Theorem of Algebra that there exists $c,\alpha_i\in\Bbb C$ such that
$$p(x)=c(x-\alpha_1)\cdots(x-\alpha_n),$$

i.e. we can factor $p$ in linear terms.

Now, what about polynomials $p(x,y)$ in two variables?

Is it still true that we can factor $p(x,y)$ in linear terms? I.e. can we always write
$$p(x,y)=c(\alpha_1x+\beta_1y+\gamma_1)\cdots(\alpha_nx+\beta_ny+\gamma_n)$$
for some $c,\alpha_i,\beta_i,\gamma_i\in\Bbb C$?