No, they haven't printed new money. For one thing, most of that money is a guarantee, not hard currency. But most of the influx of money into the US has come from foreign investment.

Also, you seem to be arguing that the way the government will avoid inflation is by creating inflation. The easiest way to cause inflation is by printing more money. This increases the amount of currency relative to the buying power of the economy, and makes the dollar worth less. Most of the sane bailout packages include things like buying mortgages at a loss or restructuring those mortgages. That keeps the loss of wealth contained within the sectors that caused the loss (or, really, bubble) to begin with. Printing up money to cover that just spreads the loss to everyone, and ends up hurting the poorer rather than the wealthier.