I can't answer your question well but I can point you toward a good reference.
Chapter 7 of Marden's Geometry of Polynomials (1949) is dedicated to bounding the zeros of a polynomial by functions of the coefficients of the polynomial. A central tool in the proofs is indeed Rouché's theorem. In chapter 9 he also considers the problem of finding the number of zeros which lie in a half-plane.
As an aside, I get a lot of use out of the Eneström-Kakeya theorem, a good reference for which is this pdf. See also this question on M.SE. One statement of the theorem goes like
If $0 \leq a_0 \leq a_1 \leq \cdots \leq a_n$ then all zeros of the polynomial
$$a_0 + a_1 z + \cdots + a_n z^n$$
lie in the closed unit disk.
In the Samuelson paper you linked he considers the polynomial
p(x) = -.027 - .013x + .220x^2 - .398 x^3 + x^4.
By the K-E theorem the related polynomial
q(x) &= p(-x) + .027 \\
&= .013x + .220 x^2 + .398x^3 + x^4
has all its roots in $|x| \leq 1$, and thus so does $p(x) + .027$. Now on the circle $|x| = 1$ we have
|p(x) + .027| \geq \left|x^4\right| - \left(|-.398|\left|x^3\right| + |.220|\left|x^2\right| + |-.013||x|\right) = .369 > |-.027|
by the triangle inequality, so by Rouché's theorem we conclude that $p(x)$ has all its roots in $|x| < 1$.