Why couldn't Euler extend his method of solving a quartic to solve a quintic polynomial?

Question

Euler creates a structure/identity (which are now generalised and known as Newton's identities) to solve a depressed quartic equation, where the coefficients are essentialy that of a cubic. We know by Newton's identies we can extended this pattern for a depressed quintic in terms of elementary symmetric functions. It should look something like this $$p^5+q^5+u^5+v^5= x^5 -5(pq+pu+pv+qu+qv+uv)x^3.... $$ where $x = p+q+u+v$. The question is: what was eulers attempts/insights in all of this?