There has been growing interest in high-order tensor methods for nonconvex optimization in machine learning as these methods provide better/optimal worst-case evaluation complexity, stability to parameter tuning, and robustness to problem conditioning. The well-known pth-order adaptive regularization (ARp) method relies crucially on repeatedly minimising a nonconvex multivariate Taylor-based polynomial sub-problem. It remains an open question to find efficient techniques to minimise such a sub-problem for p>3.In this paper, we propose a second-order method (SQO) for the AR3 (ARp with p=3) sub-problem. SQO approximates the special-structure quartic polynomial sub-problem from above and below by using second-order models that can be minimised efficiently and globally.We prove that SQO finds a local minimiser of a quartic polynomial, but in practice, due to its construction, it can find a much lower minimum than cubic regularization approaches. This encourages us to continue our quest for algorithmic techniques that find approximately global solutions for such polynomials.