How to factor 3*(x^4) + 4*(x^3) - 3

I know it has two x-intercepts but I'm not sure how to factor polynomials with degrees greater than 2..

It doesn't factor completely over the integers. You can factor it as but that's about all you can do.

There's no way I can find the exact values of the zeroes of this function algebraically?

Both real roots are irrational. There is a quartic formula, but I wouldn't bother memorizing it. Newton's method is probably the best "algebraic" way of solving it.

Let . For some initial "guess" , let and, for , . As gets larger, you should approach a root of f(x).