All those subscripts are confusing, so temporarily,

let x1=y, x2=z, then the problem becomes:

We first find f(y+k, z) by replacing y by (y+k) 

f(y+k, z) = a + b(y+k) + cz + d(y+k)z

Put the z by the d in the last term to make
it easier to distribute:

f(y+k, z) = a + b(y+k) + cz + dz(y+k)

Distribute:

f(y+k, z) = a + by + bk + cz + dzy + dzk

Get the last two terms in alphabetical order:

f(y+k, z) = a + by + bk + cz + dyz + dkz

Now we just put the equation for f(y, z) underneath 
with like terms in line, and subtract:

          f(y+k, z) = a + by + bk + cz + dyz + dkz
            f(y, z) = a + by +      cz + dyz 
--------------------------------------------------
f(y+k, z) - f(y, z) =          bk            + dkz    

Four of the terms cancel, so we have

f(y+k,z) - f(y,z) =  bk + dkz

Put back the confusing subscripts:  y=x1, z=x2,

f(x1+k, x2) - f(x1, x2) = bk + dkx2

Edwin