Drop your jaw moderately. Touch the tongue tip behind the bottom front teeth and lift the mid-front part slightly toward the roof.

Americans pronounce exploration as ehks-pluh-RAY-shuhn (/ˌɛkspləˈreɪʃən/). Stress falls on the third syllable — keep everything else short and quick. You'll hear it in sentences like "Space exploration pushes the boundaries of human knowledge".
Record yourself saying "exploration" and play it back. The mic stays on your device — nothing's uploaded.
4 syllables, 11 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
Drop your jaw moderately. Touch the tongue tip behind the bottom front teeth and lift the mid-front part slightly toward the roof.

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

Place your tongue tip near the roof of your mouth behind your top teeth. Push air through the narrow gap. No voicing.

Press your lips together to stop the air, then release. No vocal cord vibration.

Place the tip of your tongue against the alveolar ridge just behind your top front teeth, the same contact point as /t/, /d/, and /n/. The difference is what happens to the air: for /l/, you let it flow continuously around the <em>sides</em> of the tongue (that's why /l/ is called a lateral). Turn your voice on the whole time. Lips stay relaxed, no rounding or flaring. For the Dark L variant at the end of a syllable, also pull the back of the tongue up and back toward the soft palate.

Relax your lips, jaw, and tongue completely. Drop your jaw slightly and keep the tongue neutral.
The schwa before R disappears — R becomes the vowel of the syllable. This is the 'er' sound without a distinct vowel before it.

Start with your jaw slightly open and the front of your tongue forward and slightly up. Glide upward, your jaw closes a little more and your tongue arches higher toward the roof of the mouth.
Flare your lips and lift the mid-front tongue close to the roof of your mouth. Blow air through without voicing.

Relax your lips, jaw, and tongue completely. Drop your jaw slightly and keep the tongue neutral.
The schwa before N disappears — N becomes the vowel of the syllable. Go straight from the previous consonant to N.

The textbook way isn't wrong — it's just not how anyone actually says it.
In "exploration", the short unstressed vowel before "n" disappears — the schwa is absorbed and the "n" becomes the syllable nucleus on its own. Schwa is absorbed — consonant becomes the syllable nucleus.
Stress falls on the third syllable, not the others. Stretch RAY — 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.