Lorentz transformation derivation (probably simple calculus)

Technically this is for physics but I think my problem is weak calculus so I'd appreciate some help please.

This is what I have in my notes:

$c^2dt^2-dx^2=c^2dt'^2-dx'^2$

Then using the following;

$t'=Dx+Et[br]x'=Ax+Bt[br]y'=y[br]z'=z[br]$

somehow the next line reads
$c^2dt^2-dx^2=c^2(D^2dx^2+E^2dt^2+2DEdxdt)-(A^2dx^2+B^2dt^2+2ABdxdt)$

I follow the rest of the derivation but basically don't get how

$dt'^2=D^2dx^2+E^2dt^2+2DEdxdt$
and
$dx'^2=A^2dx^2+B^2dt^2+2ABdxdt$

Is there some differentiation procedure I'm forgetting for differentiating:
$t'=Dx+Et[br]x'=Ax+Bt$ ?

Thanks.