In this paper, we study static and spherically symmetric black hole (BH) solutions in the scalar-tensor theories with the coupling of the scalar field to the Gauss-Bonnet (GB) term $xi (phi) R_{rm GB}$, where $R_{rm GB}:=R^2-4R^{alphabeta}R_{alphabeta}+R^{alphabetamu u}R_{alphabetamu u}$ is the GB invariant and $xi(phi)$ is a function of the scalar field $phi$. Recently, it was shown that in these theories scalarized static and spherically symmetric BH solutions which are different from the Schwarzschild solution and possess the nontrivial profiles of the scalar field can be realized for certain choices of the coupling functions and parameters. These scalarized BH solutions are classified in terms of the number of nodes of the scalar field. It was then pointed out that in the case of the pure quadratic order coupling to the GB term, $xi(phi)=eta phi^2/8$, scalarized BH solutions with any number of nodes are unstable against the radial perturbation. In order to see how a higher order power of $phi$ in the coupling function $xi(phi)$ affects the properties of the scalarized BHs and their stability, we investigate scalarized BH solutions in the presence of the quartic order term in the GB coupling function, $xi(phi)=eta phi^2 (1+alpha phi^2)/8$. We clarify that the existence of the higher order term in the coupling function can realize scalarized BHs with zero nodes of the scalar field which are stable against the radial perturbation.
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