Press your lips together. Air flows through your nose. Vocal cords vibrate.

Americans pronounce manufacturer as man-yuh-FAK-cher-er (/ˌmænjəˈfæktʃərər/). In "manufacturer", the "a" vowel before M or N raises and fronts toward [eə] — the tongue pulls up and forward, breaking the vowel into a tense glide as it anticipates the nasal. This is called the Cat-Vowel Before M/N, a hallmark of natural-sounding American speech. It comes out as MAN·yuh·FAK·cher·er. Stress falls on the third syllable — keep everything else short and quick. You'll hear it in sentences like "The smartphone manufacturer announced groundbreaking features yesterday".
Record yourself saying "manufacturer" and play it back. The mic stays on your device — nothing's uploaded.
5 syllables, 11 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
Press your lips together. Air flows through your nose. Vocal cords vibrate.

The tongue relaxes down in the back and the corners of the lips relax before the consonant. This adds a schwa-like 'uh' relaxation after the /æ/. Think of it as 'relaxing out of the vowel' — it is no longer a pure /æ/ sound.

Touch the tip or front edge of your tongue to the roof of your mouth behind your teeth. Air flows through your nose.

Lift the middle of your tongue toward the roof of your mouth, but stop just short of touching. /j/ is an approximant, not a stop. The tongue tip stays down, lightly resting near the back of your bottom front teeth. Voice runs through the whole gesture, and the tongue glides smoothly down into the next vowel. The lips stay neutral or pre-shape for the upcoming vowel (rounding early for OO in <em>youth</em>, for example).

Relax your lips, jaw, and tongue completely. Drop your jaw slightly and keep the tongue neutral.
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.

Click any sentence to see the full breakdown — every link, every reduction, every flap-T.
The textbook way isn't wrong — it's just not how anyone actually says it.
In "manufacturer", the "a" vowel before M or N raises and fronts toward [eə] — the tongue pulls up and forward, breaking the vowel into a tense glide as it anticipates the nasal. The "/æ/" vowel raises and fronts before M or N — tongue pulls up and forward, producing a tense [eə] glide (between /e/ and /ə/). Not a pure /æ/.
In "manufacturer", the "k" 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 third 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.