Finding a quartic with integer coefficients such that $\sqrt{3} −\sqrt{5}$ is a zero Find integers $a,b,c,d$ and $e$ such that $\sqrt{3} −\sqrt{5}$ is a solution to the equation:
$$ax^4 + bx^3 + cx^2 + dx + e = 0.$$
Being new to quartic equations I wasn't sure how to solve this problem, must we use the quartic formula?
Any help is greatly appreciated!
Thanks

Since $$\sum_{cyc}(2a^2b^2-a^4)=(a+b+c)(a+b-c)(a+c-b)(b+c-a),$$
we have that $\sqrt3-\sqrt5$ is a root of the equation
$$2(3x^2+5x^2+15)-x^4-9-25=0$$ or
$$x^4-16x^2+4=0$$
