Factor the polynomial:

3x²-y²+z²-2xy-4xz

We try for a factorization of
the form (Ax+By+Cz)(Dx+Ey+Fz)

We observe:

3x² can only be factored 
feasibly as 3x and x, so if 

3x²-y²+z²-2xy-4xz 

factors at all, it would 
have to factor this way:

(3x_y_z)(x_y_z)

with the proper signs in the blanks:

The term -y² can only be factored 
feasibly as y and -y.

The term z² can only be factored 
feasibly either as either 
z and z, or -z and -z.

The terms in xy will be _3xy_xy,
and since that must have sum -2xy,
it can only be -3xy+xy, so we know 
that the factorization thus far
must be:

(3x+y_z)(x-y_z)

The terms in xz will be _3xz_xz,
and since that sum must be -4xz,
it must be -3xz-xz, so we know that
the factorization must be

(3x+y-z)(x-y-z)

This will be correct if it turns out
that the yz terms cancel out since
there are no yz terms in the original
polynomial.  And it does turn out that
the terms in yz are -yz+yz and they
do cancel out.

So the factorization is

(3x+y-z)(x-y-z)

Edwin