By definition of covariance,
[tex]\mathrm{Cov}(X,Y)=\mathbb E[(X-\mathbb E[X])(Y-\mathbb E[Y])][/tex]
[tex]\mathrm{Cov}(X,Y)=\mathbb E[XY-\mathbb E[X]Y-X\mathbb E[Y]+\mathbb E[X]\mathbb E[Y]]=\mathbb E[XY]-\mathbb E[X]\mathbb E[Y][/tex]
We have
[tex]\mathbb E[(aX-b)(cY-d)]=\mathbb E[acXY-adX-bcY+bd][/tex]
[tex]=ac\mathbb E[XY]-ad\mathbb E[X]-bc\mathbb E[Y]+bd[/tex]
[tex]\mathbb E[aX-b]=a\mathbb E[X]-b[/tex]
[tex]\mathbb E[cY-d]=c\mathbb E[Y]-d[/tex]
[tex]\mathbb E[aX-b]\mathbb E[cY-d]=ac\mathbb E[X]\mathbb E[Y]-ad\mathbb E[X]-bc\mathbb E[Y]+bd[/tex]
Putting everything together, we find the covariance reduces to
[tex]\mathrm{Cov}(aX-b,cY-d)=ac(\mathbb E[XY]-\mathbb E[X]\mathbb E[Y])=ac\mathrm{Cov}(X,Y)[/tex]
as desired.
