French school form ran out, beginning to exclaim "Five gold rings!" (3,7)
I believe the answer is:
art nouveau
'french school' is the definition.
(I've seen this before)
'form ran out beginning to exclaim five gold rings' is the wordplay.
'form' is an anagram indicator (another form of the letters).
'beginning to' indicates taking the first letters.
'five' becomes 'v' (Roman numeral).
'gold' becomes 'Au' (Au is the chemical symbol for gold).
'rings' means one lot of letters goes inside another (some letters are ringed or contained by others).
The first letter of 'exclaim' is 'e'.
'v'+'au'='vau'
'e' going within 'vau' is 'veau'.
'ran'+'out'='ranout'
'ranout' is an anagram of 'artnou'.
'artnou'+'veau'='ART NOUVEAU'
(Other definitions for art nouveau that I've seen before include "Beardsley's stuff" , "design movement" , "Decorative style (of flowing curves, floral motifs)" , "no longer in style" , "French school of painting and architecture of the 1880s" .)