[panick21_]

Yes. See:

https://en.wikipedia.org/wiki/SABRE_(rocket_engine)

https://en.wikipedia.org/wiki/Skylon_(spacecraft)

But there are lot of problems with this and it adds a huge amount of complexity.

[phkahler]

Maybe too small an effect, but what if they put 8 NACA scoops on the sides of Falcon 9 and just fed the air in between the center and outer engines. It would get accelerated out with all the regular exhaust, and I suppose would cause a small back pressure on the center engine in particular, but they could throttle it slightly if that were a problem.

[mhio]

How would feeding air outside of an engine increase performance?

Unless the gas is accelerated inside the engine bell, I don't see a force being applied to the rocket.

[addaon]

This approach is often called a thrust augmentor, e.g. https://doi.org/10.1023/B:FLUI.0000045678.92653.98. Basically you're using excess energy in a high-speed jet exhaust to accelerate ambient air, increasing the total momentum at a lower exhaust speed.

[trhway]

That was the approach on a USSR N1 Moon rocket. And also on a small air-defense missile Gnom.