I am stuck with this elementary algebra problem.It asks to find the factorization of the following quartic polynomial.
$x^4-3x-2$
Clearly this polynomial contains no rational root by Rational Root Theorem. From the graph plotted in $Desmos$ I saw that it contains two real roots.Then this polynomial can be factored non-trivially in two quadratic polynomials. But what is the factorization? I failed to find it.
Hint Since the polynomial is monic and has integer coefficients, any quadratic factorization into rational polynomials must be of the form $$x^4 - 3 x - 2 = (x^2 + a_1 x + a_0)(x^2 + b_1 x + b_0).$$ Expanding and comparing constant terms gives $a_0 b_0 = -2$, so one of $a_0, b_0$ is $\pm 2$ and the other is (respectively) $\mp 1$. By relabeling, we may as well take $a_0 = \pm 2$. For each of those possibilities, try to solve for $a_1, b_1$.