There is no expression for a root of the general polynomial equation of degree 5 in terms of its coefficients using addition, subtraction, multiplication, division and root extraction. More precisely (Galois): the symmetric group S₅ is not solvable — its composition series does not terminate in cyclic factors.
Cardano's school solved cubics (1545, Tartaglia/Ferro scandal included) and quartics followed immediately; 250 years of attempts on the quintic produced Lagrange's 1770 systematic resolvent analysis and Ruffini's 1799 incomplete impossibility proof. Évariste Galois, dead at 20 in 1832, reframed everything through the group of symmetries of roots — his manuscripts rejected, lost, and finally vindicated posthumously by Liouville in 1846.
Impossibility proven by Niels Henrik Abel, 1824 (self-published, six pages, at his own expense); the precise criterion — solvable Galois group — by Galois (~1830). Some specific quintics ARE solvable in radicals (x⁵−2=0); the general one is not, because S₅ hides the simple group A₅. The birth of abstract group theory, and mathematics' first great lesson that asking WHY a method fails beats hunting for the formula.