You explained it pretty well. I'm not sure where you're having trouble, so I'll just give a few hints/suggestions:

1) I like to de-clutter such problems. I'd set u = x/L and k = PL^3/(180ei), then work the problem for f(u) = k(3u^5 - 10 u^3 + 7u).

Since 0<=x<=L, have 0<=u<=1. (In the end, of course, will have to convert back to x and use L = 3 m.)

2) Sometimes a 4th degree equation is a quadratic equation in a square power. i.e. z^4 - 6z^2 - 10 = 0 is solveable

It's (z^2)^2 - 6(z^2) - 10 = 0, so if w = z^2, it's w^2 - 6w - 10 = 0, so w = [-(-6) +- sqrt(36-(-40)]/2 = 3+- sqrt(19).

Then z = +- sqrt[ 3+ sqrt(19)] and z = +- sqrt[sqrt(19)-3] i

Can you say where you're having difficulty with this problem?