Solve always tries to give you explicit formulas for the solutions to equations. However, it is a basic mathematical result that, for sufficiently complicated equations, explicit algebraic formulas in terms of radicals cannot be given. If you have an algebraic equation in one variable, and the highest power of the variable is at most four, then the Wolfram Language can always give you formulas for the solutions. However, if the highest power is five or more, it may be mathematically impossible to give explicit algebraic formulas for all the solutions.
