After asking this question and getting a hint for a solution in the answer, I wish to ask the following question:
How to find all irreducible polynomials in two variables over $\mathbb{C}$, having (total) degree two?
A partial answer: Write $u=ax^2+bxy+cy^2+dx+ey+f$, where $a,b,c,d,e,f \in \mathbb{C}$.
Then $u=ax^2+dx+ey+f$ is an example for an irreducible polynomial, by a sufficient criterion which is brought in one of the answers to this question.
Remark: This question is relevant, and perhaps answers my current question. However, I prefer 'a pure algebraic' answer instead of an 'algebraic geometry' answer; is it true that considering partial derivatives of $u$ answers my question and how?