My laptop has started randomly adding spaces. I went into the settings and turned off key repeat. I also cleaned the keyboard. As you can see, no help. I’m just starting a Masters program and this is not good.
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1st question: space bar. First thing to do to work toward a solution is see if a second keyboard does the same thing. If you have a wired keyboard, test it first then let me know how it reacts. You could also pair a bluetooth keyboard and try as well. This will rule out or confirm either software or hardware being the issue.

I plugged in an older keyboard and it seemed to work fine. Of note, I also cleaned my keyboard (without a ton of expertise) and it didn’t correct the issue. What would you suggest?
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Well if an external keyboard works fine then that rules out any software issue and places blame pretty confidently on the onboard keyboard having a physical problem. Try some compressed air to hopefully dislodge anything causing the issue physically. There is a setting for sticky keys within the Accessibility controls. Imagine say a person with health concerns that has a hard time controlling their motor skills and hits a key multiple times due to shaking. There is a way for the keyboard to ignore the second entry so you don’t accidentally double stroke on keys. That could be a work around for you as well but ultimately it sounds like it’s probably a hardware issue that Apple would, at worst, have to replace the keyboard. AppleCare would cover that if it’s within it’s 3 year period (given you purchased AppleCare). First year, it’s covered under warranty regardless.