Lift your bottom lip to touch the very bottom of your top front teeth. Blow air through this contact point without voicing.

Americans pronounce factory as FAK-tuh-ree (/ˈfæktəri/). Stress falls on the first syllable — keep everything else short and quick. You'll hear it in sentences like "The factory was fined for releasing toxic chemicals into the air".
Record yourself saying "factory" and play it back. The mic stays on your device — nothing's uploaded.
3 syllables, 7 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
Lift your bottom lip to touch the very bottom of your top front teeth. Blow air through this contact point without voicing.

Drop the jaw noticeably. Keep the body of the tongue low and forward, and don't let the back of the tongue raise toward the soft palate. Pull the lip corners back slightly, almost a starting smile.

Raise the back of your tongue to touch the soft palate (velum). Stop the air, then release.

The textbook way isn't wrong — it's just not how anyone actually says it.
In "factory", the "t" is not released — the articulators get into position but hold without the burst of air. Air stops but there's no release burst — the articulators hold position.
Stress falls on the first syllable, not the others. Stretch FAK — keep everything else short and quick.
Don't pronounce the first syllable too fully. The unstressed syllable reduces to a schwa — the lazy "uh" sound — in casual speech.