There is a general procedure to solve quartic equations. The first step is to depress the polynomial, i.e. translate the variable to cancel the cubic term. Your polynomial is already in depressed form.
Then you try a factorization of the form
which is built to yield a depressed polynomial.
By identification with $x^4+1$,
which easily gives
In more general cases (more nonzero coefficients), the procedure leads to a cubic equation, for which there is also a systematic procedure. More at https://en.wikipedia.org/wiki/Quartic_function#Solution_methods.