fidn all solutions to
x^4 + 3x^2 = 4x^3 + 1
The way to do this is to rearrange things so that you have a polynomial = 0. Then if you can factor it, the product of all the factors is zero only if one of the factors is zero.
x^4 - 4x^3 + 3x^2 - 1 = 0
Now, there's no easy way to solve a quartic, so look for easy-to-check roots.
f(1) = -1 nope
f(-1) = 7 nope
Since the only possible rational roots are 1 or -1. we're stuck. A handy dandy polynomial grapher (at solvemymath . com) shows that the roots are about -1.4 and 3.1.
There are two complex roots. Makes me think there's a mistake in the polynomial as given.