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

Americans pronounce scare as SKAIR (/skɛr/). It's 3 sounds in 1 syllable. The R curls back and blends into the vowel — they sound as one continuous unit.
Record yourself saying "scare" and play it back. The mic stays on your device — nothing's uploaded.
1 syllable, 3 sounds. Explore each sound's mouth shape and how it's made.
Place your tongue tip near the roof of your mouth behind your top teeth. Push air through the narrow gap. No voicing.

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

Start with the 'eh' vowel mouth position. Pull the tongue back and up while flaring the lips for the 'r'.
The textbook way isn't wrong — it's just not how anyone actually says it.
Americans use a relaxed retroflex R — the tongue curls back rather than rolling. The R is one continuous sound with the vowel before it, not two separate sounds.