Why did Eddie Mars blackmailed Rutledge instead of general Sternwood?

The General points out that his hold on life is slim. The idea is Mars was blackmailing Vivian. He'd cash in the moment that General Sternwood dies. He was smart, he was biding his time with his meal ticket.

Basically Carmen killed Regan when she was having a jealous fit (Because Sean or Rusty in the book didn't like her). Carmen, acting like a little girl, informed Vivian what she did. Between her drinking (and drugs?) she forgot about it or blocked it out. Mars buried the body in an oil sump. So Mars hid the body, and arranged an alibi that Regan ran off with Mrs. Mars.

In the book, Marlowe takes Carmen out back mansion near some old oil derricks, to teach her how to shoot. She has another jealous fit (She knows Marlowe loves Vivian not her) takes a series of shots at Marlowe in anger, but the gun is filled with blanks. She flips out. Marlowe explains to Vivian that Carmen needs to be taken somewhere to get treated/professional help.

The title of the story comes from his closing. "Where does it matter where you're buried when you're sleeping the big sleep."

The Big Sleep is basically death.

Vivian Rutledge is an easier mark -- paying people off with daddy's money is about all she knows how to do, so its just an alternative means of tapping Sternwood's money. Plus she's inclined to keep it quiet to not burden daddy, and its likely there's a decent history of buying Carmen out of trouble.

The risk of tapping Sternwood is that what happened will happen -- he'll seek outside help from a hardboiled PI to suppress whatever blackmail scheme is going on and discourage them from further schemes.

In this era, though, I'd actually expect Sternwood to have the money and connections to get a bent but loyal LAPD detective involved. This gives you a lot more oomph on the "discouraging future blackmail schemes" angle.