There is a general formula for solving quadratic equations, namely the Quadratic Formula.

For third degree equations of the form $ax^3+bx^2+cx+d=0$, there is a set of thee equations: one for each root.

Is there a general formula for solving equations of the form $ax^4+bx^3+cx^2+dx+e=0$ ?

How about for higher degrees? If not, why not?