When the federal reserve prints money you, as a commoner, don't see a dime of it. The federal reserve only transacts new capital (because its not just raw bank notes, its financial securities in general) into a select cartel of national banks.
Those banks, being private enterprise, are owned largely by the same class that owns everything else. If you own shares of the S&P 500 you own pieces of about ~two dozen banks. These are largely the banks that get treasury money which they then loan out at rates often fixed by the federal reserve in exchange for basically free money.
And the trick with that money is that the bank isn't lending philanthropically, its lending to make a profit. So using the current environment as an example you aren't getting favorable bailout loan conditions from Bank of America just because the treasury gave them a hundred billion dollars to doll out, major corporations that want to use that nigh-free money to buy back stock or buy out competition get it.
There are several branches on this tree of money flows but the economics of the last 50 years and particularly this depression from the botched COVID response only point towards the disconnect between the economy of the working poor and the economy of the owner class that gets to see the stock market hit all time record highs while the fed finally agrees to $600 stimulus checks when 50 million people are 6 months behind on rent.
And if you want to see this mechanism in action in simplest terms, the fed bought 3 trillion in securities and the stock market recovered at about 30% rates to hit all time highs again after crashing in March. That kind of recovery is unprecedented and unheard of, but because of direct constant cash flow out of the fed all the investment panic dried up because free capital was flooding investors coffers to keep buying back in with. The dozen richest men all collectively became upwards of 50% richer this year as a direct consequence. Inflation won't come close to those gains.