There is an easy method of determining whether a monic polynomial
$$\sum_0^n a_k x^k$$
with all $a_k \in \Bbb{Z}$ and $a_n = 1$ has any integer roots.
At least it is easy if you can factor the integer $a_0$ -- you just try $\pm$ all possible factors of $a_0$.
There is an easy method of determining if a non-monic polynomial has any rational root. For example, $$ 30x^3 -7x^2 -7x + 2$$
has roots $$\left\{\frac25, \frac13, -\frac12 \right\}$$
You can find these by trying very fraction $\pm \frac{p}{q}$ where $p$ divides $\lvert a_0\rvert$ and $q$ divides $\lvert a_n\rvert$. It takes a bit longer than for integer roots, but it is relatively inspiration-free (other than the factoring, again). In both cases, the method either finds a root or shows there are none.
But is there an method of determining if a polynomial over $\Bbb{Z}$ has any quadratic factors with rational coefficients (and finding one of those factors)? For example,
$$x^6 - 16x^5 + 87 x^4 -468 x^3 + 2057 x^2 -2955 x + 240$$
has a factor $(x^2-15x+48)$ and one can determine what that the constant term had to be a factor of $240$. But is there an easy rule of thumb for what to try for the linear term?
In other words, is there an algorithm involving trying out a finite number of potential factors, to determine if one of them works?