Prime is able to solve your fourth order equation but its not able to display the result because it feels that its too large to display.

But you can assign the result a function in the four input arguments.
Once these arguments have values assigned you can evaluate this function. You will get all four solutions. Prime can't know if the results are real or not or which are positive or negative without knowing specific values for w1,E,I and L.

But I wonder why you would bother with symbolic calculations at all. If at the end you are just looking for numeric results its better to use Prime's numeric facilities - like the "root" function:
