Lipschitz-like behaviour of quartic polynomials I have observed the following phenomenon: 
Let the biquadratic $q(x)=x^4-Ax^2+B$ have four real roots and perturb it by a linear factor $p(x)=q(x)+mx$, so that $m$ not too large with respect to $A,B$. 
Then the roots of $p(x)$ (assuming they are real) are very close to the roots of $q(x)$. 
Example:
$q(x)=x^4-15x^2+20$ has roots $\pm 1.22,\pm 3.68$. The perturbation $q(x)-4x$ has roots $-3.5,-1.4,1.065,3.83$ (I rounded to 1-2 digits for clarity).
The difference between corresponding roots of $q(x)$ and $r(x)$ does not exceed 0.185 in this case, which I consider quite close (good enough for my purposes).

Is there an explanation for this phenomenon?

I found two relevant papers, however one seems to have uncspecified constants $C$ and another goes into deep theory. I am looking for a simple, usable, result or at least for a simple explanation, if there is one.

You should look at that source of all wisdom: Kato's Perturbation theory of linear operators, which, in the first chapter, discusses the perturbation of eigenvalues of a matrix (before plunging into infinite dimensions in subsequent sections). Apply that discussion to the companion matrix of your polynomial.
Of course, for quartics, there is an explicit formula in radicals, so you can write down the perturbation as explicitly as you like using that.
