Eventually Paolo Ruffini and Niels Abel were able to prove that there is no quintic formula. This is somewhat surprising; even though the zeroes exist, there is no single finite expression of +, -, ×, ÷, and radicals that can produce them from the coefficients for all quintics. (One can resort to infinite expressions; Newton's method provides one. See also ‘limit of a sequence’.)
But their proof did not generalise to higher degrees. The honour of proving the quartic formula to be the last of its kind, ie there was no sextic, septic, octic, formula, and so on, fell to Evariste Galois, who had an ingenious insight which reduced the issue to an important but solved question of group theory.