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2 The question changed slightly.; added 2 characters in body

I am looking for factorization of polynomials of several variables in the way outlined below.

Consider a second degree polynomial of two variables over the complex numbers.

P(x,y)

"P(x,y) = Ax^2 + Bxy + Cy^2 + Dx + Ey + F" (see the edit below)

Experimenting with some polynomials of this sort showed me that factorization is possible in the following way.

P(x,y)

"P(x,y) = (ax + by + c)(dx + ey + ff)"  (see the edit below) ,

the coefficients being over the complex numbers.

So, given an nth degree polynomial in n variables without a constant term, is it always possible to factorize it into n linear factors each having  n variables in the above way? (This rings  bells  about the fundamental theorem of algebra) algebra).

[EDIT: I am sorry, I erred. I have edited my question. The edit is that the polynomial has no constant term:
P(x,y) = Ax^2 + Bxy + Cy^2 + Dx + Ey
and in the expected factorization, the last linear factor does not have a constant term, too:
P(x,y) = (ax + by + c)(dx + ey + f) ]

I kept the original question as it is for documentation purposes.


 
 
 
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Factorizing polynomials of several variables (in a different perespective)

I am looking for factorization of polynomials of several variables in the way outlined below.

Consider a second degree polynomial of two variables over the complex numbers.

P(x,y) = Ax^2 + Bxy + Cy^2 + Dx + Ey + F


Experimenting with some polynomials of this sort showed me that factorization is possible in the following way.

P(x,y) = (ax + by + c)(dx + ey + f) , the coefficients being over the complex numbers.

So, given an nth degree polynomial in n variables, is it always possible to factorize it into n linear factors each having n variables? (This rings bells about the fundamental theorem of algebra)