For fixed degree monic polynomials with real coefficients, the direct and inverse problems of root inclusion are denned. The direct problem is to find the infimal region Γ1γof the C-plane that contains all roots of the bounded coefficient norm polynomials. The inverse problem is to define the maximal subregion Γ2γof Γ1γsuch that all polynomials with their roots in Γ2γare bounded norm coefficient polynomials. Those two problems are solved in the case of stable and totally unstable polynomials and the boundaries of Γ1γ,Γ2γ: regions are defined as branches of an algebraic function defined by the norm bound y. For general polynomials, the inverse problem is also solved for polynomials of any degree, whereas the direct problem is discussed in the case of third-order polynomials.