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

Americans pronounce finances as FAHY-nan-suhz (/ˈfaɪnænsəz/). In "finances", 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, and it's why Americans sound more relaxed than the textbook. It comes out as FAHY·nan·suhz. Stress falls on the first syllable — keep everything else short and quick. You'll hear it in sentences like "She has a valid power of attorney to manage his finances".
Record yourself saying "finances" and play it back. The mic stays on your device — nothing's uploaded.
3 syllables, 8 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
Touch the tip or front edge of your tongue to the roof of your mouth behind your teeth. Air flows through your nose.

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.

The textbook way isn't wrong — it's just not how anyone actually says it.
In "finances", 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/.
Stress falls on the first syllable, not the others. Stretch FAHY — keep everything else short and quick.
Don't pronounce the second syllable too fully. The unstressed syllable reduces to a schwa — the lazy "uh" sound — in casual speech.